↑

Properties

Label 1160.2.a.e.1.3
Level $1160$
Weight $2$
Character 1160.1
Self dual yes
Analytic conductor $9.263$
Analytic rank $1$
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] = [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,-2,0,-3,0,2] 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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(2.17009\) 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)
 
\(f(q)\) \(=\) \(q+1.70928 q^{3} -1.00000 q^{5} -0.630898 q^{7} -0.0783777 q^{9} -3.70928 q^{11} -3.07838 q^{13} -1.70928 q^{15} -6.04945 q^{17} -1.36910 q^{19} -1.07838 q^{21} +1.70928 q^{23} +1.00000 q^{25} -5.26180 q^{27} +1.00000 q^{29} -4.78765 q^{31} -6.34017 q^{33} +0.630898 q^{35} -2.63090 q^{37} -5.26180 q^{39} +8.34017 q^{41} +1.12783 q^{43} +0.0783777 q^{45} +10.3896 q^{47} -6.60197 q^{49} -10.3402 q^{51} +9.02052 q^{53} +3.70928 q^{55} -2.34017 q^{57} +5.75872 q^{59} -9.60197 q^{61} +0.0494483 q^{63} +3.07838 q^{65} -8.44748 q^{67} +2.92162 q^{69} -9.75872 q^{71} -4.29072 q^{73} +1.70928 q^{75} +2.34017 q^{77} +6.23287 q^{79} -8.75872 q^{81} -11.1545 q^{83} +6.04945 q^{85} +1.70928 q^{87} +2.58145 q^{89} +1.94214 q^{91} -8.18342 q^{93} +1.36910 q^{95} +13.6514 q^{97} +0.290725 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} - 3 q^{5} + 2 q^{7} + 3 q^{9} - 4 q^{11} - 6 q^{13} + 2 q^{15} - 8 q^{19} - 2 q^{23} + 3 q^{25} - 8 q^{27} + 3 q^{29} - 4 q^{31} - 8 q^{33} - 2 q^{35} - 4 q^{37} - 8 q^{39} + 14 q^{41} - 18 q^{43}+ \cdots + 8 q^{99}+O(q^{100}) \) 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
\(3\) 1.70928 0.986851 0.493425 − 0.869788i \(-0.335745\pi\)
0.493425 + 0.869788i \(0.335745\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −0.630898 −0.238457 −0.119228 − 0.992867i \(-0.538042\pi\)
−0.119228 + 0.992867i \(0.538042\pi\)
\(8\) 0 0
\(9\) −0.0783777 −0.0261259
\(10\) 0 0
\(11\) −3.70928 −1.11839 −0.559194 − 0.829037i \(-0.688889\pi\)
−0.559194 + 0.829037i \(0.688889\pi\)
\(12\) 0 0
\(13\) −3.07838 −0.853788 −0.426894 − 0.904302i \(-0.640392\pi\)
−0.426894 + 0.904302i \(0.640392\pi\)
\(14\) 0 0
\(15\) −1.70928 −0.441333
\(16\) 0 0
\(17\) −6.04945 −1.46721 −0.733603 − 0.679578i \(-0.762163\pi\)
−0.733603 + 0.679578i \(0.762163\pi\)
\(18\) 0 0
\(19\) −1.36910 −0.314094 −0.157047 − 0.987591i \(-0.550197\pi\)
−0.157047 + 0.987591i \(0.550197\pi\)
\(20\) 0 0
\(21\) −1.07838 −0.235321
\(22\) 0 0
\(23\) 1.70928 0.356409 0.178204 − 0.983994i \(-0.442971\pi\)
0.178204 + 0.983994i \(0.442971\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.26180 −1.01263
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −4.78765 −0.859888 −0.429944 − 0.902856i \(-0.641467\pi\)
−0.429944 + 0.902856i \(0.641467\pi\)
\(32\) 0 0
\(33\) −6.34017 −1.10368
\(34\) 0 0
\(35\) 0.630898 0.106641
\(36\) 0 0
\(37\) −2.63090 −0.432517 −0.216258 − 0.976336i \(-0.569385\pi\)
−0.216258 + 0.976336i \(0.569385\pi\)
\(38\) 0 0
\(39\) −5.26180 −0.842562
\(40\) 0 0
\(41\) 8.34017 1.30252 0.651258 − 0.758856i \(-0.274242\pi\)
0.651258 + 0.758856i \(0.274242\pi\)
\(42\) 0 0
\(43\) 1.12783 0.171992 0.0859959 − 0.996295i \(-0.472593\pi\)
0.0859959 + 0.996295i \(0.472593\pi\)
\(44\) 0 0
\(45\) 0.0783777 0.0116839
\(46\) 0 0
\(47\) 10.3896 1.51548 0.757741 − 0.652555i \(-0.226303\pi\)
0.757741 + 0.652555i \(0.226303\pi\)
\(48\) 0 0
\(49\) −6.60197 −0.943138
\(50\) 0 0
\(51\) −10.3402 −1.44791
\(52\) 0 0
\(53\) 9.02052 1.23906 0.619532 − 0.784972i \(-0.287322\pi\)
0.619532 + 0.784972i \(0.287322\pi\)
\(54\) 0 0
\(55\) 3.70928 0.500159
\(56\) 0 0
\(57\) −2.34017 −0.309963
\(58\) 0 0
\(59\) 5.75872 0.749722 0.374861 − 0.927081i \(-0.377690\pi\)
0.374861 + 0.927081i \(0.377690\pi\)
\(60\) 0 0
\(61\) −9.60197 −1.22941 −0.614703 − 0.788759i \(-0.710724\pi\)
−0.614703 + 0.788759i \(0.710724\pi\)
\(62\) 0 0
\(63\) 0.0494483 0.00622990
\(64\) 0 0
\(65\) 3.07838 0.381826
\(66\) 0 0
\(67\) −8.44748 −1.03202 −0.516012 − 0.856581i \(-0.672584\pi\)
−0.516012 + 0.856581i \(0.672584\pi\)
\(68\) 0 0
\(69\) 2.92162 0.351722
\(70\) 0 0
\(71\) −9.75872 −1.15815 −0.579074 − 0.815275i \(-0.696586\pi\)
−0.579074 + 0.815275i \(0.696586\pi\)
\(72\) 0 0
\(73\) −4.29072 −0.502191 −0.251096 − 0.967962i \(-0.580791\pi\)
−0.251096 + 0.967962i \(0.580791\pi\)
\(74\) 0 0
\(75\) 1.70928 0.197370
\(76\) 0 0
\(77\) 2.34017 0.266687
\(78\) 0 0
\(79\) 6.23287 0.701252 0.350626 − 0.936516i \(-0.385969\pi\)
0.350626 + 0.936516i \(0.385969\pi\)
\(80\) 0 0
\(81\) −8.75872 −0.973192
\(82\) 0 0
\(83\) −11.1545 −1.22436 −0.612182 − 0.790717i \(-0.709708\pi\)
−0.612182 + 0.790717i \(0.709708\pi\)
\(84\) 0 0
\(85\) 6.04945 0.656155
\(86\) 0 0
\(87\) 1.70928 0.183254
\(88\) 0 0
\(89\) 2.58145 0.273633 0.136817 − 0.990596i \(-0.456313\pi\)
0.136817 + 0.990596i \(0.456313\pi\)
\(90\) 0 0
\(91\) 1.94214 0.203592
\(92\) 0 0
\(93\) −8.18342 −0.848581
\(94\) 0 0
\(95\) 1.36910 0.140467
\(96\) 0 0
\(97\) 13.6514 1.38609 0.693046 − 0.720894i \(-0.256268\pi\)
0.693046 + 0.720894i \(0.256268\pi\)
\(98\) 0 0
\(99\) 0.290725 0.0292189
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.e.1.3 ✓ 3
4.3 odd 2 2320.2.a.r.1.1 3
5.4 even 2 5800.2.a.r.1.1 3
8.3 odd 2 9280.2.a.bl.1.3 3
8.5 even 2 9280.2.a.bv.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1160.2.a.e.1.3 ✓ 3 1.1 even 1 trivial
2320.2.a.r.1.1 3 4.3 odd 2
5800.2.a.r.1.1 3 5.4 even 2
9280.2.a.bl.1.3 3 8.3 odd 2
9280.2.a.bv.1.1 3 8.5 even 2