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Properties

Label 2160.2.by.b.289.1
Level $2160$
Weight $2$
Character 2160.289
Analytic conductor $17.248$
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] = [2160,2,Mod(289,2160)] 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("2160.289"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2160, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 2160 = 2^{4} \cdot 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2160.by (of order \(6\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,0,0,0,-2,0,0,0,0,0,4] 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: no
Analytic conductor: \(17.2476868366\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 90)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 289.1
Root \(0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 2160.289
Dual form 2160.2.by.b.1009.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.23205 + 0.133975i) q^{5} +(0.866025 + 0.500000i) q^{7} +(1.00000 - 1.73205i) q^{11} +(-5.19615 + 3.00000i) q^{13} -2.00000i q^{17} +6.00000 q^{19} +(0.866025 - 0.500000i) q^{23} +(4.96410 - 0.598076i) q^{25} +(-4.50000 + 7.79423i) q^{29} +(-1.00000 - 1.73205i) q^{31} +(-2.00000 - 1.00000i) q^{35} -2.00000i q^{37} +(-5.50000 - 9.52628i) q^{41} +(-3.46410 - 2.00000i) q^{43} +(-6.06218 - 3.50000i) q^{47} +(-3.00000 - 5.19615i) q^{49} +(-2.00000 + 4.00000i) q^{55} +(-2.00000 - 3.46410i) q^{59} +(3.50000 - 6.06218i) q^{61} +(11.1962 - 7.39230i) q^{65} +(-9.52628 + 5.50000i) q^{67} -6.00000 q^{71} -4.00000i q^{73} +(1.73205 - 1.00000i) q^{77} +(6.00000 - 10.3923i) q^{79} +(9.52628 + 5.50000i) q^{83} +(0.267949 + 4.46410i) q^{85} +1.00000 q^{89} -6.00000 q^{91} +(-13.3923 + 0.803848i) q^{95} +(-6.92820 - 4.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{5} + 4 q^{11} + 24 q^{19} + 6 q^{25} - 18 q^{29} - 4 q^{31} - 8 q^{35} - 22 q^{41} - 12 q^{49} - 8 q^{55} - 8 q^{59} + 14 q^{61} + 24 q^{65} - 24 q^{71} + 24 q^{79} + 8 q^{85} + 4 q^{89} - 24 q^{91}+ \cdots - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2160\mathbb{Z}\right)^\times\).

\(n\) \(271\) \(1297\) \(1621\) \(2081\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(5\) −2.23205 + 0.133975i −0.998203 + 0.0599153i
\(6\) 0 0
\(7\) 0.866025 + 0.500000i 0.327327 + 0.188982i 0.654654 − 0.755929i \(-0.272814\pi\)
−0.327327 + 0.944911i \(0.606148\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 − 1.73205i 0.301511 − 0.522233i −0.674967 − 0.737848i \(-0.735842\pi\)
0.976478 + 0.215615i \(0.0691756\pi\)
\(12\) 0 0
\(13\) −5.19615 + 3.00000i −1.44115 + 0.832050i −0.997927 − 0.0643593i \(-0.979500\pi\)
−0.443227 + 0.896410i \(0.646166\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 2.00000i − 0.485071i −0.970143 − 0.242536i \(-0.922021\pi\)
0.970143 − 0.242536i \(-0.0779791\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 − 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.866025 − 0.500000i 0.180579 − 0.104257i −0.406986 − 0.913434i \(-0.633420\pi\)
0.587565 + 0.809177i \(0.300087\pi\)
\(24\) 0 0
\(25\) 4.96410 − 0.598076i 0.992820 − 0.119615i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.50000 + 7.79423i −0.835629 + 1.44735i 0.0578882 + 0.998323i \(0.481563\pi\)
−0.893517 + 0.449029i \(0.851770\pi\)
\(30\) 0 0
\(31\) −1.00000 − 1.73205i −0.179605 − 0.311086i 0.762140 − 0.647412i \(-0.224149\pi\)
−0.941745 + 0.336327i \(0.890815\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.00000 − 1.00000i −0.338062 − 0.169031i
\(36\) 0 0
\(37\) − 2.00000i − 0.328798i −0.986394 − 0.164399i \(-0.947432\pi\)
0.986394 − 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −5.50000 − 9.52628i −0.858956 − 1.48775i −0.872926 − 0.487852i \(-0.837780\pi\)
0.0139704 − 0.999902i \(-0.495553\pi\)
\(42\) 0 0
\(43\) −3.46410 − 2.00000i −0.528271 − 0.304997i 0.212041 − 0.977261i \(-0.431989\pi\)
−0.740312 + 0.672264i \(0.765322\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.06218 − 3.50000i −0.884260 − 0.510527i −0.0121990 − 0.999926i \(-0.503883\pi\)
−0.872060 + 0.489398i \(0.837217\pi\)
\(48\) 0 0
\(49\) −3.00000 − 5.19615i −0.428571 − 0.742307i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) −2.00000 + 4.00000i −0.269680 + 0.539360i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.00000 − 3.46410i −0.260378 − 0.450988i 0.705965 − 0.708247i \(-0.250514\pi\)
−0.966342 + 0.257260i \(0.917180\pi\)
\(60\) 0 0
\(61\) 3.50000 − 6.06218i 0.448129 − 0.776182i −0.550135 − 0.835076i \(-0.685424\pi\)
0.998264 + 0.0588933i \(0.0187572\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 11.1962 − 7.39230i 1.38871 − 0.916903i
\(66\) 0 0
\(67\) −9.52628 + 5.50000i −1.16382 + 0.671932i −0.952217 − 0.305424i \(-0.901202\pi\)
−0.211604 + 0.977356i \(0.567869\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) − 4.00000i − 0.468165i −0.972217 − 0.234082i \(-0.924791\pi\)
0.972217 − 0.234082i \(-0.0752085\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.73205 − 1.00000i 0.197386 − 0.113961i
\(78\) 0 0
\(79\) 6.00000 − 10.3923i 0.675053 − 1.16923i −0.301401 − 0.953498i \(-0.597454\pi\)
0.976453 − 0.215728i \(-0.0692125\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 9.52628 + 5.50000i 1.04565 + 0.603703i 0.921427 − 0.388552i \(-0.127024\pi\)
0.124218 + 0.992255i \(0.460358\pi\)
\(84\) 0 0
\(85\) 0.267949 + 4.46410i 0.0290632 + 0.484200i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.00000 0.106000 0.0529999 − 0.998595i \(-0.483122\pi\)
0.0529999 + 0.998595i \(0.483122\pi\)
\(90\) 0 0
\(91\) −6.00000 −0.628971
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −13.3923 + 0.803848i −1.37402 + 0.0824730i
\(96\) 0 0
\(97\) −6.92820 − 4.00000i −0.703452 − 0.406138i 0.105180 − 0.994453i \(-0.466458\pi\)
−0.808632 + 0.588315i \(0.799792\pi\)
\(98\) 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 2160.2.by.b.289.1 4
3.2 odd 2 720.2.by.a.529.2 4
4.3 odd 2 270.2.i.a.19.1 4
5.4 even 2 inner 2160.2.by.b.289.2 4
9.4 even 3 inner 2160.2.by.b.1009.2 4
9.5 odd 6 720.2.by.a.49.1 4
12.11 even 2 90.2.i.a.79.2 yes 4
15.14 odd 2 720.2.by.a.529.1 4
20.3 even 4 1350.2.e.a.451.1 2
20.7 even 4 1350.2.e.i.451.1 2
20.19 odd 2 270.2.i.a.19.2 4
36.7 odd 6 810.2.c.c.649.2 2
36.11 even 6 810.2.c.b.649.1 2
36.23 even 6 90.2.i.a.49.1 ✓ 4
36.31 odd 6 270.2.i.a.199.2 4
45.4 even 6 inner 2160.2.by.b.1009.1 4
45.14 odd 6 720.2.by.a.49.2 4
60.23 odd 4 450.2.e.g.151.1 2
60.47 odd 4 450.2.e.b.151.1 2
60.59 even 2 90.2.i.a.79.1 yes 4
180.7 even 12 4050.2.a.g.1.1 1
180.23 odd 12 450.2.e.g.301.1 2
180.43 even 12 4050.2.a.be.1.1 1
180.47 odd 12 4050.2.a.x.1.1 1
180.59 even 6 90.2.i.a.49.2 yes 4
180.67 even 12 1350.2.e.i.901.1 2
180.79 odd 6 810.2.c.c.649.1 2
180.83 odd 12 4050.2.a.j.1.1 1
180.103 even 12 1350.2.e.a.901.1 2
180.119 even 6 810.2.c.b.649.2 2
180.139 odd 6 270.2.i.a.199.1 4
180.167 odd 12 450.2.e.b.301.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
90.2.i.a.49.1 ✓ 4 36.23 even 6
90.2.i.a.49.2 yes 4 180.59 even 6
90.2.i.a.79.1 yes 4 60.59 even 2
90.2.i.a.79.2 yes 4 12.11 even 2
270.2.i.a.19.1 4 4.3 odd 2
270.2.i.a.19.2 4 20.19 odd 2
270.2.i.a.199.1 4 180.139 odd 6
270.2.i.a.199.2 4 36.31 odd 6
450.2.e.b.151.1 2 60.47 odd 4
450.2.e.b.301.1 2 180.167 odd 12
450.2.e.g.151.1 2 60.23 odd 4
450.2.e.g.301.1 2 180.23 odd 12
720.2.by.a.49.1 4 9.5 odd 6
720.2.by.a.49.2 4 45.14 odd 6
720.2.by.a.529.1 4 15.14 odd 2
720.2.by.a.529.2 4 3.2 odd 2
810.2.c.b.649.1 2 36.11 even 6
810.2.c.b.649.2 2 180.119 even 6
810.2.c.c.649.1 2 180.79 odd 6
810.2.c.c.649.2 2 36.7 odd 6
1350.2.e.a.451.1 2 20.3 even 4
1350.2.e.a.901.1 2 180.103 even 12
1350.2.e.i.451.1 2 20.7 even 4
1350.2.e.i.901.1 2 180.67 even 12
2160.2.by.b.289.1 4 1.1 even 1 trivial
2160.2.by.b.289.2 4 5.4 even 2 inner
2160.2.by.b.1009.1 4 45.4 even 6 inner
2160.2.by.b.1009.2 4 9.4 even 3 inner
4050.2.a.g.1.1 1 180.7 even 12
4050.2.a.j.1.1 1 180.83 odd 12
4050.2.a.x.1.1 1 180.47 odd 12
4050.2.a.be.1.1 1 180.43 even 12