Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1620,4,Mod(541,1620)] 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("1620.541"); S:= CuspForms(chi, 4); 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, 4, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1620.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,5,0,-32] 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: no
Analytic conductor: \(95.5830942093\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 541.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1620.541
Dual form 1620.4.i.g.1081.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(2.50000 + 4.33013i) q^{5} +(-16.0000 + 27.7128i) q^{7} +(18.0000 - 31.1769i) q^{11} +(5.00000 + 8.66025i) q^{13} +78.0000 q^{17} +140.000 q^{19} +(-96.0000 - 166.277i) q^{23} +(-12.5000 + 21.6506i) q^{25} +(3.00000 - 5.19615i) q^{29} +(8.00000 + 13.8564i) q^{31} -160.000 q^{35} -34.0000 q^{37} +(-195.000 - 337.750i) q^{41} +(26.0000 - 45.0333i) q^{43} +(204.000 - 353.338i) q^{47} +(-340.500 - 589.763i) q^{49} +114.000 q^{53} +180.000 q^{55} +(258.000 + 446.869i) q^{59} +(29.0000 - 50.2295i) q^{61} +(-25.0000 + 43.3013i) q^{65} +(446.000 + 772.495i) q^{67} +120.000 q^{71} -646.000 q^{73} +(576.000 + 997.661i) q^{77} +(584.000 - 1011.52i) q^{79} +(-366.000 + 633.931i) q^{83} +(195.000 + 337.750i) q^{85} +1590.00 q^{89} -320.000 q^{91} +(350.000 + 606.218i) q^{95} +(-97.0000 + 168.009i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{5} - 32 q^{7} + 36 q^{11} + 10 q^{13} + 156 q^{17} + 280 q^{19} - 192 q^{23} - 25 q^{25} + 6 q^{29} + 16 q^{31} - 320 q^{35} - 68 q^{37} - 390 q^{41} + 52 q^{43} + 408 q^{47} - 681 q^{49} + 228 q^{53}+ \cdots - 194 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(811\) \(1297\) \(1541\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(5\) 2.50000 + 4.33013i 0.223607 + 0.387298i
\(6\) 0 0
\(7\) −16.0000 + 27.7128i −0.863919 + 1.49635i 0.00419795 + 0.999991i \(0.498664\pi\)
−0.868117 + 0.496360i \(0.834670\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 18.0000 − 31.1769i 0.493382 − 0.854563i −0.506589 − 0.862188i \(-0.669094\pi\)
0.999971 + 0.00762479i \(0.00242707\pi\)
\(12\) 0 0
\(13\) 5.00000 + 8.66025i 0.106673 + 0.184763i 0.914421 − 0.404765i \(-0.132647\pi\)
−0.807747 + 0.589529i \(0.799313\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 78.0000 1.11281 0.556405 − 0.830911i \(-0.312180\pi\)
0.556405 + 0.830911i \(0.312180\pi\)
\(18\) 0 0
\(19\) 140.000 1.69043 0.845216 − 0.534425i \(-0.179472\pi\)
0.845216 + 0.534425i \(0.179472\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −96.0000 − 166.277i −0.870321 − 1.50744i −0.861665 − 0.507478i \(-0.830578\pi\)
−0.00865615 − 0.999963i \(-0.502755\pi\)
\(24\) 0 0
\(25\) −12.5000 + 21.6506i −0.100000 + 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.00000 − 5.19615i 0.0192099 − 0.0332725i −0.856261 − 0.516544i \(-0.827218\pi\)
0.875471 + 0.483272i \(0.160552\pi\)
\(30\) 0 0
\(31\) 8.00000 + 13.8564i 0.0463498 + 0.0802801i 0.888270 − 0.459323i \(-0.151908\pi\)
−0.841920 + 0.539603i \(0.818574\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −160.000 −0.772712
\(36\) 0 0
\(37\) −34.0000 −0.151069 −0.0755347 − 0.997143i \(-0.524066\pi\)
−0.0755347 + 0.997143i \(0.524066\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −195.000 − 337.750i −0.742778 − 1.28653i −0.951226 − 0.308495i \(-0.900175\pi\)
0.208448 − 0.978033i \(-0.433159\pi\)
\(42\) 0 0
\(43\) 26.0000 − 45.0333i 0.0922084 − 0.159710i −0.816232 − 0.577725i \(-0.803941\pi\)
0.908440 + 0.418015i \(0.137274\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 204.000 − 353.338i 0.633116 − 1.09659i −0.353795 − 0.935323i \(-0.615109\pi\)
0.986911 − 0.161266i \(-0.0515578\pi\)
\(48\) 0 0
\(49\) −340.500 − 589.763i −0.992711 − 1.71943i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 114.000 0.295455 0.147727 − 0.989028i \(-0.452804\pi\)
0.147727 + 0.989028i \(0.452804\pi\)
\(54\) 0 0
\(55\) 180.000 0.441294
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 258.000 + 446.869i 0.569301 + 0.986058i 0.996635 + 0.0819641i \(0.0261193\pi\)
−0.427335 + 0.904094i \(0.640547\pi\)
\(60\) 0 0
\(61\) 29.0000 − 50.2295i 0.0608700 − 0.105430i −0.833985 − 0.551788i \(-0.813946\pi\)
0.894855 + 0.446358i \(0.147279\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −25.0000 + 43.3013i −0.0477057 + 0.0826286i
\(66\) 0 0
\(67\) 446.000 + 772.495i 0.813247 + 1.40859i 0.910580 + 0.413334i \(0.135636\pi\)
−0.0973322 + 0.995252i \(0.531031\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 120.000 0.200583 0.100291 − 0.994958i \(-0.468022\pi\)
0.100291 + 0.994958i \(0.468022\pi\)
\(72\) 0 0
\(73\) −646.000 −1.03573 −0.517867 − 0.855461i \(-0.673274\pi\)
−0.517867 + 0.855461i \(0.673274\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 576.000 + 997.661i 0.852484 + 1.47655i
\(78\) 0 0
\(79\) 584.000 − 1011.52i 0.831711 − 1.44056i −0.0649702 − 0.997887i \(-0.520695\pi\)
0.896681 − 0.442678i \(-0.145971\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −366.000 + 633.931i −0.484021 + 0.838348i −0.999832 − 0.0183540i \(-0.994157\pi\)
0.515811 + 0.856703i \(0.327491\pi\)
\(84\) 0 0
\(85\) 195.000 + 337.750i 0.248832 + 0.430990i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1590.00 1.89370 0.946852 − 0.321669i \(-0.104244\pi\)
0.946852 + 0.321669i \(0.104244\pi\)
\(90\) 0 0
\(91\) −320.000 −0.368628
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 350.000 + 606.218i 0.377992 + 0.654701i
\(96\) 0 0
\(97\) −97.0000 + 168.009i −0.101535 + 0.175863i −0.912317 − 0.409484i \(-0.865709\pi\)
0.810782 + 0.585348i \(0.199042\pi\)
\(98\) 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 1620.4.i.g.541.1 2
3.2 odd 2 1620.4.i.a.541.1 2
9.2 odd 6 60.4.a.b.1.1 ✓ 1
9.4 even 3 inner 1620.4.i.g.1081.1 2
9.5 odd 6 1620.4.i.a.1081.1 2
9.7 even 3 180.4.a.c.1.1 1
36.7 odd 6 720.4.a.c.1.1 1
36.11 even 6 240.4.a.j.1.1 1
45.2 even 12 300.4.d.d.49.2 2
45.7 odd 12 900.4.d.b.649.2 2
45.29 odd 6 300.4.a.e.1.1 1
45.34 even 6 900.4.a.b.1.1 1
45.38 even 12 300.4.d.d.49.1 2
45.43 odd 12 900.4.d.b.649.1 2
72.11 even 6 960.4.a.a.1.1 1
72.29 odd 6 960.4.a.bb.1.1 1
180.47 odd 12 1200.4.f.e.49.1 2
180.83 odd 12 1200.4.f.e.49.2 2
180.119 even 6 1200.4.a.s.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.4.a.b.1.1 ✓ 1 9.2 odd 6
180.4.a.c.1.1 1 9.7 even 3
240.4.a.j.1.1 1 36.11 even 6
300.4.a.e.1.1 1 45.29 odd 6
300.4.d.d.49.1 2 45.38 even 12
300.4.d.d.49.2 2 45.2 even 12
720.4.a.c.1.1 1 36.7 odd 6
900.4.a.b.1.1 1 45.34 even 6
900.4.d.b.649.1 2 45.43 odd 12
900.4.d.b.649.2 2 45.7 odd 12
960.4.a.a.1.1 1 72.11 even 6
960.4.a.bb.1.1 1 72.29 odd 6
1200.4.a.s.1.1 1 180.119 even 6
1200.4.f.e.49.1 2 180.47 odd 12
1200.4.f.e.49.2 2 180.83 odd 12
1620.4.i.a.541.1 2 3.2 odd 2
1620.4.i.a.1081.1 2 9.5 odd 6
1620.4.i.g.541.1 2 1.1 even 1 trivial
1620.4.i.g.1081.1 2 9.4 even 3 inner