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Properties

Label 160.2.f.a.49.4
Level $160$
Weight $2$
Character 160.49
Analytic conductor $1.278$
Analytic rank $0$
Dimension $4$
Inner twists $4$

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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] = [160,2,Mod(49,160)] 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("160.49"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(160, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 160 = 2^{5} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 160.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.27760643234\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 40)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.4
Root \(0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 160.49
Dual form 160.2.f.a.49.3

$q$-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)
 
\(f(q)\) \(=\) \(q+1.41421 q^{3} +(1.41421 + 1.73205i) q^{5} -2.44949i q^{7} -1.00000 q^{9} +3.46410i q^{11} +(2.00000 + 2.44949i) q^{15} -4.89898i q^{17} -3.46410i q^{19} -3.46410i q^{21} +2.44949i q^{23} +(-1.00000 + 4.89898i) q^{25} -5.65685 q^{27} -4.00000 q^{31} +4.89898i q^{33} +(4.24264 - 3.46410i) q^{35} -8.48528 q^{37} -4.24264 q^{43} +(-1.41421 - 1.73205i) q^{45} -7.34847i q^{47} +1.00000 q^{49} -6.92820i q^{51} +5.65685 q^{53} +(-6.00000 + 4.89898i) q^{55} -4.89898i q^{57} +10.3923i q^{59} +3.46410i q^{61} +2.44949i q^{63} +4.24264 q^{67} +3.46410i q^{69} +12.0000 q^{71} -4.89898i q^{73} +(-1.41421 + 6.92820i) q^{75} +8.48528 q^{77} +4.00000 q^{79} -5.00000 q^{81} +9.89949 q^{83} +(8.48528 - 6.92820i) q^{85} +6.00000 q^{89} -5.65685 q^{93} +(6.00000 - 4.89898i) q^{95} +4.89898i q^{97} -3.46410i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9} + 8 q^{15} - 4 q^{25} - 16 q^{31} + 4 q^{49} - 24 q^{55} + 48 q^{71} + 16 q^{79} - 20 q^{81} + 24 q^{89} + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/160\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(101\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) 1.41421 0.816497 0.408248 − 0.912871i \(-0.366140\pi\)
0.408248 + 0.912871i \(0.366140\pi\)
\(4\) 0 0
\(5\) 1.41421 + 1.73205i 0.632456 + 0.774597i
\(6\) 0 0
\(7\) − 2.44949i − 0.925820i −0.886405 − 0.462910i \(-0.846805\pi\)
0.886405 − 0.462910i \(-0.153195\pi\)
\(8\) 0 0
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 3.46410i 1.04447i 0.852803 + 0.522233i \(0.174901\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 2.00000 + 2.44949i 0.516398 + 0.632456i
\(16\) 0 0
\(17\) − 4.89898i − 1.18818i −0.804400 − 0.594089i \(-0.797513\pi\)
0.804400 − 0.594089i \(-0.202487\pi\)
\(18\) 0 0
\(19\) − 3.46410i − 0.794719i −0.917663 − 0.397360i \(-0.869927\pi\)
0.917663 − 0.397360i \(-0.130073\pi\)
\(20\) 0 0
\(21\) − 3.46410i − 0.755929i
\(22\) 0 0
\(23\) 2.44949i 0.510754i 0.966842 + 0.255377i \(0.0821996\pi\)
−0.966842 + 0.255377i \(0.917800\pi\)
\(24\) 0 0
\(25\) −1.00000 + 4.89898i −0.200000 + 0.979796i
\(26\) 0 0
\(27\) −5.65685 −1.08866
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) 4.89898i 0.852803i
\(34\) 0 0
\(35\) 4.24264 − 3.46410i 0.717137 − 0.585540i
\(36\) 0 0
\(37\) −8.48528 −1.39497 −0.697486 − 0.716599i \(-0.745698\pi\)
−0.697486 + 0.716599i \(0.745698\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −4.24264 −0.646997 −0.323498 − 0.946229i \(-0.604859\pi\)
−0.323498 + 0.946229i \(0.604859\pi\)
\(44\) 0 0
\(45\) −1.41421 − 1.73205i −0.210819 − 0.258199i
\(46\) 0 0
\(47\) − 7.34847i − 1.07188i −0.844255 − 0.535942i \(-0.819956\pi\)
0.844255 − 0.535942i \(-0.180044\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) − 6.92820i − 0.970143i
\(52\) 0 0
\(53\) 5.65685 0.777029 0.388514 − 0.921443i \(-0.372988\pi\)
0.388514 + 0.921443i \(0.372988\pi\)
\(54\) 0 0
\(55\) −6.00000 + 4.89898i −0.809040 + 0.660578i
\(56\) 0 0
\(57\) − 4.89898i − 0.648886i
\(58\) 0 0
\(59\) 10.3923i 1.35296i 0.736460 + 0.676481i \(0.236496\pi\)
−0.736460 + 0.676481i \(0.763504\pi\)
\(60\) 0 0
\(61\) 3.46410i 0.443533i 0.975100 + 0.221766i \(0.0711822\pi\)
−0.975100 + 0.221766i \(0.928818\pi\)
\(62\) 0 0
\(63\) 2.44949i 0.308607i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.24264 0.518321 0.259161 − 0.965834i \(-0.416554\pi\)
0.259161 + 0.965834i \(0.416554\pi\)
\(68\) 0 0
\(69\) 3.46410i 0.417029i
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) − 4.89898i − 0.573382i −0.958023 − 0.286691i \(-0.907445\pi\)
0.958023 − 0.286691i \(-0.0925553\pi\)
\(74\) 0 0
\(75\) −1.41421 + 6.92820i −0.163299 + 0.800000i
\(76\) 0 0
\(77\) 8.48528 0.966988
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) −5.00000 −0.555556
\(82\) 0 0
\(83\) 9.89949 1.08661 0.543305 − 0.839535i \(-0.317173\pi\)
0.543305 + 0.839535i \(0.317173\pi\)
\(84\) 0 0
\(85\) 8.48528 − 6.92820i 0.920358 − 0.751469i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 − 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −5.65685 −0.586588
\(94\) 0 0
\(95\) 6.00000 − 4.89898i 0.615587 − 0.502625i
\(96\) 0 0
\(97\) 4.89898i 0.497416i 0.968579 + 0.248708i \(0.0800060\pi\)
−0.968579 + 0.248708i \(0.919994\pi\)
\(98\) 0 0
\(99\) − 3.46410i − 0.348155i
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 160.2.f.a.49.4 4
3.2 odd 2 1440.2.d.c.1009.1 4
4.3 odd 2 40.2.f.a.29.3 yes 4
5.2 odd 4 800.2.d.f.401.2 4
5.3 odd 4 800.2.d.f.401.3 4
5.4 even 2 inner 160.2.f.a.49.2 4
8.3 odd 2 40.2.f.a.29.1 ✓ 4
8.5 even 2 inner 160.2.f.a.49.1 4
12.11 even 2 360.2.d.b.109.2 4
15.2 even 4 7200.2.k.l.3601.3 4
15.8 even 4 7200.2.k.l.3601.1 4
15.14 odd 2 1440.2.d.c.1009.3 4
16.3 odd 4 1280.2.c.i.769.2 4
16.5 even 4 1280.2.c.k.769.1 4
16.11 odd 4 1280.2.c.i.769.3 4
16.13 even 4 1280.2.c.k.769.4 4
20.3 even 4 200.2.d.e.101.1 4
20.7 even 4 200.2.d.e.101.4 4
20.19 odd 2 40.2.f.a.29.2 yes 4
24.5 odd 2 1440.2.d.c.1009.4 4
24.11 even 2 360.2.d.b.109.4 4
40.3 even 4 200.2.d.e.101.2 4
40.13 odd 4 800.2.d.f.401.1 4
40.19 odd 2 40.2.f.a.29.4 yes 4
40.27 even 4 200.2.d.e.101.3 4
40.29 even 2 inner 160.2.f.a.49.3 4
40.37 odd 4 800.2.d.f.401.4 4
60.23 odd 4 1800.2.k.m.901.4 4
60.47 odd 4 1800.2.k.m.901.1 4
60.59 even 2 360.2.d.b.109.3 4
80.3 even 4 6400.2.a.co.1.3 4
80.13 odd 4 6400.2.a.cm.1.2 4
80.19 odd 4 1280.2.c.i.769.4 4
80.27 even 4 6400.2.a.co.1.4 4
80.29 even 4 1280.2.c.k.769.2 4
80.37 odd 4 6400.2.a.cm.1.1 4
80.43 even 4 6400.2.a.co.1.1 4
80.53 odd 4 6400.2.a.cm.1.4 4
80.59 odd 4 1280.2.c.i.769.1 4
80.67 even 4 6400.2.a.co.1.2 4
80.69 even 4 1280.2.c.k.769.3 4
80.77 odd 4 6400.2.a.cm.1.3 4
120.29 odd 2 1440.2.d.c.1009.2 4
120.53 even 4 7200.2.k.l.3601.2 4
120.59 even 2 360.2.d.b.109.1 4
120.77 even 4 7200.2.k.l.3601.4 4
120.83 odd 4 1800.2.k.m.901.3 4
120.107 odd 4 1800.2.k.m.901.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.2.f.a.29.1 ✓ 4 8.3 odd 2
40.2.f.a.29.2 yes 4 20.19 odd 2
40.2.f.a.29.3 yes 4 4.3 odd 2
40.2.f.a.29.4 yes 4 40.19 odd 2
160.2.f.a.49.1 4 8.5 even 2 inner
160.2.f.a.49.2 4 5.4 even 2 inner
160.2.f.a.49.3 4 40.29 even 2 inner
160.2.f.a.49.4 4 1.1 even 1 trivial
200.2.d.e.101.1 4 20.3 even 4
200.2.d.e.101.2 4 40.3 even 4
200.2.d.e.101.3 4 40.27 even 4
200.2.d.e.101.4 4 20.7 even 4
360.2.d.b.109.1 4 120.59 even 2
360.2.d.b.109.2 4 12.11 even 2
360.2.d.b.109.3 4 60.59 even 2
360.2.d.b.109.4 4 24.11 even 2
800.2.d.f.401.1 4 40.13 odd 4
800.2.d.f.401.2 4 5.2 odd 4
800.2.d.f.401.3 4 5.3 odd 4
800.2.d.f.401.4 4 40.37 odd 4
1280.2.c.i.769.1 4 80.59 odd 4
1280.2.c.i.769.2 4 16.3 odd 4
1280.2.c.i.769.3 4 16.11 odd 4
1280.2.c.i.769.4 4 80.19 odd 4
1280.2.c.k.769.1 4 16.5 even 4
1280.2.c.k.769.2 4 80.29 even 4
1280.2.c.k.769.3 4 80.69 even 4
1280.2.c.k.769.4 4 16.13 even 4
1440.2.d.c.1009.1 4 3.2 odd 2
1440.2.d.c.1009.2 4 120.29 odd 2
1440.2.d.c.1009.3 4 15.14 odd 2
1440.2.d.c.1009.4 4 24.5 odd 2
1800.2.k.m.901.1 4 60.47 odd 4
1800.2.k.m.901.2 4 120.107 odd 4
1800.2.k.m.901.3 4 120.83 odd 4
1800.2.k.m.901.4 4 60.23 odd 4
6400.2.a.cm.1.1 4 80.37 odd 4
6400.2.a.cm.1.2 4 80.13 odd 4
6400.2.a.cm.1.3 4 80.77 odd 4
6400.2.a.cm.1.4 4 80.53 odd 4
6400.2.a.co.1.1 4 80.43 even 4
6400.2.a.co.1.2 4 80.67 even 4
6400.2.a.co.1.3 4 80.3 even 4
6400.2.a.co.1.4 4 80.27 even 4
7200.2.k.l.3601.1 4 15.8 even 4
7200.2.k.l.3601.2 4 120.53 even 4
7200.2.k.l.3601.3 4 15.2 even 4
7200.2.k.l.3601.4 4 120.77 even 4