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Properties

Label 1160.2.a.g.1.2
Level $1160$
Weight $2$
Character 1160.1
Self dual yes
Analytic conductor $9.263$
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] = [1160,2,Mod(1,1160)] 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("1160.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1160, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1160 = 2^{3} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1160.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,0,1,0,-3,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(9.26264663447\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.229.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 4x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.86081\) of defining polynomial
Character \(\chi\) \(=\) 1160.1

$q$-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)
 
\(f(q)\) \(=\) \(q-0.462598 q^{3} -1.00000 q^{5} -2.86081 q^{7} -2.78600 q^{9} +5.72161 q^{11} +0.462598 q^{13} +0.462598 q^{15} +1.93561 q^{17} -8.36842 q^{19} +1.32340 q^{21} +8.18421 q^{23} +1.00000 q^{25} +2.67660 q^{27} -1.00000 q^{29} +3.78600 q^{31} -2.64681 q^{33} +2.86081 q^{35} +3.07480 q^{37} -0.213997 q^{39} +3.72161 q^{41} +9.78600 q^{43} +2.78600 q^{45} +8.64681 q^{47} +1.18421 q^{49} -0.895410 q^{51} -2.58242 q^{53} -5.72161 q^{55} +3.87122 q^{57} +11.7562 q^{59} -9.62743 q^{61} +7.97021 q^{63} -0.462598 q^{65} +13.2936 q^{67} -3.78600 q^{69} -5.29362 q^{71} +9.63640 q^{73} -0.462598 q^{75} -16.3684 q^{77} -5.90582 q^{79} +7.11982 q^{81} +4.92520 q^{83} -1.93561 q^{85} +0.462598 q^{87} -5.07480 q^{89} -1.32340 q^{91} -1.75140 q^{93} +8.36842 q^{95} -1.66618 q^{97} -15.9404 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{3} - 3 q^{5} - 3 q^{7} + 2 q^{9} + 6 q^{11} - q^{13} - q^{15} + 5 q^{17} + 2 q^{19} - 4 q^{21} + 11 q^{23} + 3 q^{25} + 16 q^{27} - 3 q^{29} + q^{31} + 8 q^{33} + 3 q^{35} + 14 q^{37} - 11 q^{39}+ \cdots - 9 q^{97}+O(q^{100}) \) 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
\(3\) −0.462598 −0.267081 −0.133541 − 0.991043i \(-0.542635\pi\)
−0.133541 + 0.991043i \(0.542635\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −2.86081 −1.08128 −0.540641 − 0.841253i \(-0.681818\pi\)
−0.540641 + 0.841253i \(0.681818\pi\)
\(8\) 0 0
\(9\) −2.78600 −0.928668
\(10\) 0 0
\(11\) 5.72161 1.72513 0.862565 − 0.505946i \(-0.168856\pi\)
0.862565 + 0.505946i \(0.168856\pi\)
\(12\) 0 0
\(13\) 0.462598 0.128302 0.0641509 − 0.997940i \(-0.479566\pi\)
0.0641509 + 0.997940i \(0.479566\pi\)
\(14\) 0 0
\(15\) 0.462598 0.119442
\(16\) 0 0
\(17\) 1.93561 0.469454 0.234727 − 0.972061i \(-0.424580\pi\)
0.234727 + 0.972061i \(0.424580\pi\)
\(18\) 0 0
\(19\) −8.36842 −1.91985 −0.959924 − 0.280262i \(-0.909579\pi\)
−0.959924 + 0.280262i \(0.909579\pi\)
\(20\) 0 0
\(21\) 1.32340 0.288790
\(22\) 0 0
\(23\) 8.18421 1.70653 0.853263 − 0.521481i \(-0.174620\pi\)
0.853263 + 0.521481i \(0.174620\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 2.67660 0.515111
\(28\) 0 0
\(29\) −1.00000 −0.185695
\(30\) 0 0
\(31\) 3.78600 0.679986 0.339993 − 0.940428i \(-0.389575\pi\)
0.339993 + 0.940428i \(0.389575\pi\)
\(32\) 0 0
\(33\) −2.64681 −0.460750
\(34\) 0 0
\(35\) 2.86081 0.483564
\(36\) 0 0
\(37\) 3.07480 0.505495 0.252747 − 0.967532i \(-0.418666\pi\)
0.252747 + 0.967532i \(0.418666\pi\)
\(38\) 0 0
\(39\) −0.213997 −0.0342670
\(40\) 0 0
\(41\) 3.72161 0.581218 0.290609 − 0.956842i \(-0.406142\pi\)
0.290609 + 0.956842i \(0.406142\pi\)
\(42\) 0 0
\(43\) 9.78600 1.49235 0.746176 − 0.665749i \(-0.231888\pi\)
0.746176 + 0.665749i \(0.231888\pi\)
\(44\) 0 0
\(45\) 2.78600 0.415313
\(46\) 0 0
\(47\) 8.64681 1.26127 0.630633 − 0.776081i \(-0.282795\pi\)
0.630633 + 0.776081i \(0.282795\pi\)
\(48\) 0 0
\(49\) 1.18421 0.169173
\(50\) 0 0
\(51\) −0.895410 −0.125382
\(52\) 0 0
\(53\) −2.58242 −0.354722 −0.177361 − 0.984146i \(-0.556756\pi\)
−0.177361 + 0.984146i \(0.556756\pi\)
\(54\) 0 0
\(55\) −5.72161 −0.771502
\(56\) 0 0
\(57\) 3.87122 0.512755
\(58\) 0 0
\(59\) 11.7562 1.53053 0.765264 − 0.643716i \(-0.222608\pi\)
0.765264 + 0.643716i \(0.222608\pi\)
\(60\) 0 0
\(61\) −9.62743 −1.23267 −0.616333 − 0.787485i \(-0.711383\pi\)
−0.616333 + 0.787485i \(0.711383\pi\)
\(62\) 0 0
\(63\) 7.97021 1.00415
\(64\) 0 0
\(65\) −0.462598 −0.0573783
\(66\) 0 0
\(67\) 13.2936 1.62407 0.812037 − 0.583606i \(-0.198359\pi\)
0.812037 + 0.583606i \(0.198359\pi\)
\(68\) 0 0
\(69\) −3.78600 −0.455781
\(70\) 0 0
\(71\) −5.29362 −0.628237 −0.314118 − 0.949384i \(-0.601709\pi\)
−0.314118 + 0.949384i \(0.601709\pi\)
\(72\) 0 0
\(73\) 9.63640 1.12785 0.563927 − 0.825824i \(-0.309290\pi\)
0.563927 + 0.825824i \(0.309290\pi\)
\(74\) 0 0
\(75\) −0.462598 −0.0534163
\(76\) 0 0
\(77\) −16.3684 −1.86535
\(78\) 0 0
\(79\) −5.90582 −0.664457 −0.332228 − 0.943199i \(-0.607800\pi\)
−0.332228 + 0.943199i \(0.607800\pi\)
\(80\) 0 0
\(81\) 7.11982 0.791091
\(82\) 0 0
\(83\) 4.92520 0.540611 0.270305 − 0.962775i \(-0.412875\pi\)
0.270305 + 0.962775i \(0.412875\pi\)
\(84\) 0 0
\(85\) −1.93561 −0.209946
\(86\) 0 0
\(87\) 0.462598 0.0495958
\(88\) 0 0
\(89\) −5.07480 −0.537928 −0.268964 − 0.963150i \(-0.586681\pi\)
−0.268964 + 0.963150i \(0.586681\pi\)
\(90\) 0 0
\(91\) −1.32340 −0.138730
\(92\) 0 0
\(93\) −1.75140 −0.181612
\(94\) 0 0
\(95\) 8.36842 0.858582
\(96\) 0 0
\(97\) −1.66618 −0.169175 −0.0845877 − 0.996416i \(-0.526957\pi\)
−0.0845877 + 0.996416i \(0.526957\pi\)
\(98\) 0 0
\(99\) −15.9404 −1.60207
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 1160.2.a.g.1.2 ✓ 3
4.3 odd 2 2320.2.a.p.1.2 3
5.4 even 2 5800.2.a.q.1.2 3
8.3 odd 2 9280.2.a.bq.1.2 3
8.5 even 2 9280.2.a.bo.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1160.2.a.g.1.2 ✓ 3 1.1 even 1 trivial
2320.2.a.p.1.2 3 4.3 odd 2
5800.2.a.q.1.2 3 5.4 even 2
9280.2.a.bo.1.2 3 8.5 even 2
9280.2.a.bq.1.2 3 8.3 odd 2