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Properties

Label 1620.2.r.d.109.1
Level $1620$
Weight $2$
Character 1620.109
Analytic conductor $12.936$
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] = [1620,2,Mod(109,1620)] 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("1620.109"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1620, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1620.r (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,-8] 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: \(12.9357651274\)
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: \( 2^{2} \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 109.1
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 1620.109
Dual form 1620.2.r.d.1189.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.23205 - 1.86603i) q^{5} +(3.46410 - 2.00000i) q^{7} +(-2.00000 - 3.46410i) q^{11} -4.00000i q^{17} +(-3.46410 - 2.00000i) q^{23} +(-1.96410 + 4.59808i) q^{25} +(3.00000 + 5.19615i) q^{29} +(-2.00000 + 3.46410i) q^{31} +(-8.00000 - 4.00000i) q^{35} -8.00000i q^{37} +(-5.00000 + 8.66025i) q^{41} +(-3.46410 + 2.00000i) q^{43} +(3.46410 - 2.00000i) q^{47} +(4.50000 - 7.79423i) q^{49} -12.0000i q^{53} +(-4.00000 + 8.00000i) q^{55} +(-2.00000 + 3.46410i) q^{59} +(-1.00000 - 1.73205i) q^{61} +(3.46410 + 2.00000i) q^{67} +8.00000i q^{73} +(-13.8564 - 8.00000i) q^{77} +(-6.00000 - 10.3923i) q^{79} +(3.46410 - 2.00000i) q^{83} +(-7.46410 + 4.92820i) q^{85} -10.0000 q^{89} +(6.92820 - 4.00000i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{5} - 8 q^{11} + 6 q^{25} + 12 q^{29} - 8 q^{31} - 32 q^{35} - 20 q^{41} + 18 q^{49} - 16 q^{55} - 8 q^{59} - 4 q^{61} - 24 q^{79} - 16 q^{85} - 40 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times\).

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{1}{3}\right)\)

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
\(5\) −1.23205 − 1.86603i −0.550990 − 0.834512i
\(6\) 0 0
\(7\) 3.46410 − 2.00000i 1.30931 − 0.755929i 0.327327 − 0.944911i \(-0.393852\pi\)
0.981981 + 0.188982i \(0.0605189\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.00000 − 3.46410i −0.603023 − 1.04447i −0.992361 − 0.123371i \(-0.960630\pi\)
0.389338 − 0.921095i \(-0.372704\pi\)
\(12\) 0 0
\(13\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 4.00000i − 0.970143i −0.874475 − 0.485071i \(-0.838794\pi\)
0.874475 − 0.485071i \(-0.161206\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.46410 − 2.00000i −0.722315 − 0.417029i 0.0932891 − 0.995639i \(-0.470262\pi\)
−0.815604 + 0.578610i \(0.803595\pi\)
\(24\) 0 0
\(25\) −1.96410 + 4.59808i −0.392820 + 0.919615i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.00000 + 5.19615i 0.557086 + 0.964901i 0.997738 + 0.0672232i \(0.0214140\pi\)
−0.440652 + 0.897678i \(0.645253\pi\)
\(30\) 0 0
\(31\) −2.00000 + 3.46410i −0.359211 + 0.622171i −0.987829 − 0.155543i \(-0.950287\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −8.00000 − 4.00000i −1.35225 − 0.676123i
\(36\) 0 0
\(37\) − 8.00000i − 1.31519i −0.753371 − 0.657596i \(-0.771573\pi\)
0.753371 − 0.657596i \(-0.228427\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −5.00000 + 8.66025i −0.780869 + 1.35250i 0.150567 + 0.988600i \(0.451890\pi\)
−0.931436 + 0.363905i \(0.881443\pi\)
\(42\) 0 0
\(43\) −3.46410 + 2.00000i −0.528271 + 0.304997i −0.740312 − 0.672264i \(-0.765322\pi\)
0.212041 + 0.977261i \(0.431989\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.46410 − 2.00000i 0.505291 − 0.291730i −0.225605 − 0.974219i \(-0.572436\pi\)
0.730896 + 0.682489i \(0.239102\pi\)
\(48\) 0 0
\(49\) 4.50000 − 7.79423i 0.642857 − 1.11346i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 12.0000i − 1.64833i −0.566352 − 0.824163i \(-0.691646\pi\)
0.566352 − 0.824163i \(-0.308354\pi\)
\(54\) 0 0
\(55\) −4.00000 + 8.00000i −0.539360 + 1.07872i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.00000 + 3.46410i −0.260378 + 0.450988i −0.966342 − 0.257260i \(-0.917180\pi\)
0.705965 + 0.708247i \(0.250514\pi\)
\(60\) 0 0
\(61\) −1.00000 − 1.73205i −0.128037 − 0.221766i 0.794879 − 0.606768i \(-0.207534\pi\)
−0.922916 + 0.385002i \(0.874201\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.46410 + 2.00000i 0.423207 + 0.244339i 0.696449 − 0.717607i \(-0.254762\pi\)
−0.273241 + 0.961946i \(0.588096\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 8.00000i 0.936329i 0.883641 + 0.468165i \(0.155085\pi\)
−0.883641 + 0.468165i \(0.844915\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −13.8564 − 8.00000i −1.57908 − 0.911685i
\(78\) 0 0
\(79\) −6.00000 − 10.3923i −0.675053 − 1.16923i −0.976453 − 0.215728i \(-0.930788\pi\)
0.301401 − 0.953498i \(-0.402546\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.46410 − 2.00000i 0.380235 − 0.219529i −0.297686 − 0.954664i \(-0.596215\pi\)
0.677920 + 0.735135i \(0.262881\pi\)
\(84\) 0 0
\(85\) −7.46410 + 4.92820i −0.809595 + 0.534539i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −10.0000 −1.06000 −0.529999 − 0.847998i \(-0.677808\pi\)
−0.529999 + 0.847998i \(0.677808\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 6.92820 − 4.00000i 0.703452 − 0.406138i −0.105180 − 0.994453i \(-0.533542\pi\)
0.808632 + 0.588315i \(0.200208\pi\)
\(98\) 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 1620.2.r.d.109.1 4
3.2 odd 2 1620.2.r.c.109.2 4
5.4 even 2 inner 1620.2.r.d.109.2 4
9.2 odd 6 1620.2.r.c.1189.1 4
9.4 even 3 180.2.d.a.109.2 2
9.5 odd 6 60.2.d.a.49.1 ✓ 2
9.7 even 3 inner 1620.2.r.d.1189.2 4
15.14 odd 2 1620.2.r.c.109.1 4
36.23 even 6 240.2.f.b.49.2 2
36.31 odd 6 720.2.f.c.289.2 2
45.4 even 6 180.2.d.a.109.1 2
45.13 odd 12 900.2.a.h.1.1 1
45.14 odd 6 60.2.d.a.49.2 yes 2
45.22 odd 12 900.2.a.a.1.1 1
45.23 even 12 300.2.a.d.1.1 1
45.29 odd 6 1620.2.r.c.1189.2 4
45.32 even 12 300.2.a.a.1.1 1
45.34 even 6 inner 1620.2.r.d.1189.1 4
63.5 even 6 2940.2.bb.e.949.1 4
63.23 odd 6 2940.2.bb.d.949.2 4
63.32 odd 6 2940.2.bb.d.1549.1 4
63.41 even 6 2940.2.k.c.589.2 2
63.59 even 6 2940.2.bb.e.1549.2 4
72.5 odd 6 960.2.f.f.769.2 2
72.13 even 6 2880.2.f.l.1729.1 2
72.59 even 6 960.2.f.c.769.1 2
72.67 odd 6 2880.2.f.p.1729.1 2
144.5 odd 12 3840.2.d.o.2689.2 2
144.59 even 12 3840.2.d.be.2689.2 2
144.77 odd 12 3840.2.d.r.2689.1 2
144.131 even 12 3840.2.d.b.2689.1 2
180.23 odd 12 1200.2.a.a.1.1 1
180.59 even 6 240.2.f.b.49.1 2
180.67 even 12 3600.2.a.bm.1.1 1
180.103 even 12 3600.2.a.d.1.1 1
180.139 odd 6 720.2.f.c.289.1 2
180.167 odd 12 1200.2.a.s.1.1 1
315.59 even 6 2940.2.bb.e.1549.1 4
315.104 even 6 2940.2.k.c.589.1 2
315.149 odd 6 2940.2.bb.d.949.1 4
315.194 even 6 2940.2.bb.e.949.2 4
315.284 odd 6 2940.2.bb.d.1549.2 4
360.59 even 6 960.2.f.c.769.2 2
360.77 even 12 4800.2.a.bn.1.1 1
360.139 odd 6 2880.2.f.p.1729.2 2
360.149 odd 6 960.2.f.f.769.1 2
360.203 odd 12 4800.2.a.bk.1.1 1
360.229 even 6 2880.2.f.l.1729.2 2
360.293 even 12 4800.2.a.bj.1.1 1
360.347 odd 12 4800.2.a.bf.1.1 1
720.59 even 12 3840.2.d.b.2689.2 2
720.149 odd 12 3840.2.d.r.2689.2 2
720.419 even 12 3840.2.d.be.2689.1 2
720.509 odd 12 3840.2.d.o.2689.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.2.d.a.49.1 ✓ 2 9.5 odd 6
60.2.d.a.49.2 yes 2 45.14 odd 6
180.2.d.a.109.1 2 45.4 even 6
180.2.d.a.109.2 2 9.4 even 3
240.2.f.b.49.1 2 180.59 even 6
240.2.f.b.49.2 2 36.23 even 6
300.2.a.a.1.1 1 45.32 even 12
300.2.a.d.1.1 1 45.23 even 12
720.2.f.c.289.1 2 180.139 odd 6
720.2.f.c.289.2 2 36.31 odd 6
900.2.a.a.1.1 1 45.22 odd 12
900.2.a.h.1.1 1 45.13 odd 12
960.2.f.c.769.1 2 72.59 even 6
960.2.f.c.769.2 2 360.59 even 6
960.2.f.f.769.1 2 360.149 odd 6
960.2.f.f.769.2 2 72.5 odd 6
1200.2.a.a.1.1 1 180.23 odd 12
1200.2.a.s.1.1 1 180.167 odd 12
1620.2.r.c.109.1 4 15.14 odd 2
1620.2.r.c.109.2 4 3.2 odd 2
1620.2.r.c.1189.1 4 9.2 odd 6
1620.2.r.c.1189.2 4 45.29 odd 6
1620.2.r.d.109.1 4 1.1 even 1 trivial
1620.2.r.d.109.2 4 5.4 even 2 inner
1620.2.r.d.1189.1 4 45.34 even 6 inner
1620.2.r.d.1189.2 4 9.7 even 3 inner
2880.2.f.l.1729.1 2 72.13 even 6
2880.2.f.l.1729.2 2 360.229 even 6
2880.2.f.p.1729.1 2 72.67 odd 6
2880.2.f.p.1729.2 2 360.139 odd 6
2940.2.k.c.589.1 2 315.104 even 6
2940.2.k.c.589.2 2 63.41 even 6
2940.2.bb.d.949.1 4 315.149 odd 6
2940.2.bb.d.949.2 4 63.23 odd 6
2940.2.bb.d.1549.1 4 63.32 odd 6
2940.2.bb.d.1549.2 4 315.284 odd 6
2940.2.bb.e.949.1 4 63.5 even 6
2940.2.bb.e.949.2 4 315.194 even 6
2940.2.bb.e.1549.1 4 315.59 even 6
2940.2.bb.e.1549.2 4 63.59 even 6
3600.2.a.d.1.1 1 180.103 even 12
3600.2.a.bm.1.1 1 180.67 even 12
3840.2.d.b.2689.1 2 144.131 even 12
3840.2.d.b.2689.2 2 720.59 even 12
3840.2.d.o.2689.1 2 720.509 odd 12
3840.2.d.o.2689.2 2 144.5 odd 12
3840.2.d.r.2689.1 2 144.77 odd 12
3840.2.d.r.2689.2 2 720.149 odd 12
3840.2.d.be.2689.1 2 720.419 even 12
3840.2.d.be.2689.2 2 144.59 even 12
4800.2.a.bf.1.1 1 360.347 odd 12
4800.2.a.bj.1.1 1 360.293 even 12
4800.2.a.bk.1.1 1 360.203 odd 12
4800.2.a.bn.1.1 1 360.77 even 12