Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3240,2,Mod(1081,3240)] 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("3240.1081"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3240, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3240.q (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,1,0,-2,0,0,0,-2,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.8715302549\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 360)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1081.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 3240.1081
Dual form 3240.2.q.n.2161.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(0.500000 - 0.866025i) q^{5} +(-1.00000 - 1.73205i) q^{7} +(-1.00000 - 1.73205i) q^{11} +(-2.00000 + 3.46410i) q^{13} -2.00000 q^{17} +4.00000 q^{19} +(-4.00000 + 6.92820i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(5.00000 + 8.66025i) q^{29} +(-2.00000 + 3.46410i) q^{31} -2.00000 q^{35} +(4.00000 + 6.92820i) q^{43} +(-4.00000 - 6.92820i) q^{47} +(1.50000 - 2.59808i) q^{49} +6.00000 q^{53} -2.00000 q^{55} +(7.00000 - 12.1244i) q^{59} +(7.00000 + 12.1244i) q^{61} +(2.00000 + 3.46410i) q^{65} +(2.00000 - 3.46410i) q^{67} +12.0000 q^{71} +6.00000 q^{73} +(-2.00000 + 3.46410i) q^{77} +(6.00000 + 10.3923i) q^{79} +(-2.00000 - 3.46410i) q^{83} +(-1.00000 + 1.73205i) q^{85} -12.0000 q^{89} +8.00000 q^{91} +(2.00000 - 3.46410i) q^{95} +(7.00000 + 12.1244i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{5} - 2 q^{7} - 2 q^{11} - 4 q^{13} - 4 q^{17} + 8 q^{19} - 8 q^{23} - q^{25} + 10 q^{29} - 4 q^{31} - 4 q^{35} + 8 q^{43} - 8 q^{47} + 3 q^{49} + 12 q^{53} - 4 q^{55} + 14 q^{59} + 14 q^{61}+ \cdots + 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times\).

\(n\) \(1297\) \(1621\) \(2431\) \(3161\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(5\) 0.500000 − 0.866025i 0.223607 − 0.387298i
\(6\) 0 0
\(7\) −1.00000 − 1.73205i −0.377964 − 0.654654i 0.612801 − 0.790237i \(-0.290043\pi\)
−0.990766 + 0.135583i \(0.956709\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 − 1.73205i −0.301511 − 0.522233i 0.674967 − 0.737848i \(-0.264158\pi\)
−0.976478 + 0.215615i \(0.930824\pi\)
\(12\) 0 0
\(13\) −2.00000 + 3.46410i −0.554700 + 0.960769i 0.443227 + 0.896410i \(0.353834\pi\)
−0.997927 + 0.0643593i \(0.979500\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 − 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.00000 + 6.92820i −0.834058 + 1.44463i 0.0607377 + 0.998154i \(0.480655\pi\)
−0.894795 + 0.446476i \(0.852679\pi\)
\(24\) 0 0
\(25\) −0.500000 − 0.866025i −0.100000 − 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.00000 + 8.66025i 0.928477 + 1.60817i 0.785872 + 0.618389i \(0.212214\pi\)
0.142605 + 0.989780i \(0.454452\pi\)
\(30\) 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
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(42\) 0 0
\(43\) 4.00000 + 6.92820i 0.609994 + 1.05654i 0.991241 + 0.132068i \(0.0421616\pi\)
−0.381246 + 0.924473i \(0.624505\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.00000 − 6.92820i −0.583460 − 1.01058i −0.995066 − 0.0992202i \(-0.968365\pi\)
0.411606 − 0.911362i \(-0.364968\pi\)
\(48\) 0 0
\(49\) 1.50000 − 2.59808i 0.214286 − 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 − 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.00000 − 12.1244i 0.911322 − 1.57846i 0.0991242 − 0.995075i \(-0.468396\pi\)
0.812198 − 0.583382i \(-0.198271\pi\)
\(60\) 0 0
\(61\) 7.00000 + 12.1244i 0.896258 + 1.55236i 0.832240 + 0.554416i \(0.187058\pi\)
0.0640184 + 0.997949i \(0.479608\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.00000 + 3.46410i 0.248069 + 0.429669i
\(66\) 0 0
\(67\) 2.00000 − 3.46410i 0.244339 − 0.423207i −0.717607 − 0.696449i \(-0.754762\pi\)
0.961946 + 0.273241i \(0.0880957\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000 1.42414 0.712069 − 0.702109i \(-0.247758\pi\)
0.712069 + 0.702109i \(0.247758\pi\)
\(72\) 0 0
\(73\) 6.00000 0.702247 0.351123 − 0.936329i \(-0.385800\pi\)
0.351123 + 0.936329i \(0.385800\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.00000 + 3.46410i −0.227921 + 0.394771i
\(78\) 0 0
\(79\) 6.00000 + 10.3923i 0.675053 + 1.16923i 0.976453 + 0.215728i \(0.0692125\pi\)
−0.301401 + 0.953498i \(0.597454\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −2.00000 − 3.46410i −0.219529 − 0.380235i 0.735135 − 0.677920i \(-0.237119\pi\)
−0.954664 + 0.297686i \(0.903785\pi\)
\(84\) 0 0
\(85\) −1.00000 + 1.73205i −0.108465 + 0.187867i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −12.0000 −1.27200 −0.635999 − 0.771690i \(-0.719412\pi\)
−0.635999 + 0.771690i \(0.719412\pi\)
\(90\) 0 0
\(91\) 8.00000 0.838628
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.00000 − 3.46410i 0.205196 − 0.355409i
\(96\) 0 0
\(97\) 7.00000 + 12.1244i 0.710742 + 1.23104i 0.964579 + 0.263795i \(0.0849741\pi\)
−0.253837 + 0.967247i \(0.581693\pi\)
\(98\) 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 3240.2.q.n.1081.1 2
3.2 odd 2 3240.2.q.d.1081.1 2
9.2 odd 6 3240.2.q.d.2161.1 2
9.4 even 3 360.2.a.c.1.1 ✓ 1
9.5 odd 6 360.2.a.d.1.1 yes 1
9.7 even 3 inner 3240.2.q.n.2161.1 2
36.23 even 6 720.2.a.i.1.1 1
36.31 odd 6 720.2.a.a.1.1 1
45.4 even 6 1800.2.a.i.1.1 1
45.13 odd 12 1800.2.f.h.649.1 2
45.14 odd 6 1800.2.a.f.1.1 1
45.22 odd 12 1800.2.f.h.649.2 2
45.23 even 12 1800.2.f.d.649.1 2
45.32 even 12 1800.2.f.d.649.2 2
72.5 odd 6 2880.2.a.n.1.1 1
72.13 even 6 2880.2.a.bd.1.1 1
72.59 even 6 2880.2.a.e.1.1 1
72.67 odd 6 2880.2.a.w.1.1 1
180.23 odd 12 3600.2.f.q.2449.2 2
180.59 even 6 3600.2.a.bh.1.1 1
180.67 even 12 3600.2.f.g.2449.1 2
180.103 even 12 3600.2.f.g.2449.2 2
180.139 odd 6 3600.2.a.bd.1.1 1
180.167 odd 12 3600.2.f.q.2449.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
360.2.a.c.1.1 ✓ 1 9.4 even 3
360.2.a.d.1.1 yes 1 9.5 odd 6
720.2.a.a.1.1 1 36.31 odd 6
720.2.a.i.1.1 1 36.23 even 6
1800.2.a.f.1.1 1 45.14 odd 6
1800.2.a.i.1.1 1 45.4 even 6
1800.2.f.d.649.1 2 45.23 even 12
1800.2.f.d.649.2 2 45.32 even 12
1800.2.f.h.649.1 2 45.13 odd 12
1800.2.f.h.649.2 2 45.22 odd 12
2880.2.a.e.1.1 1 72.59 even 6
2880.2.a.n.1.1 1 72.5 odd 6
2880.2.a.w.1.1 1 72.67 odd 6
2880.2.a.bd.1.1 1 72.13 even 6
3240.2.q.d.1081.1 2 3.2 odd 2
3240.2.q.d.2161.1 2 9.2 odd 6
3240.2.q.n.1081.1 2 1.1 even 1 trivial
3240.2.q.n.2161.1 2 9.7 even 3 inner
3600.2.a.bd.1.1 1 180.139 odd 6
3600.2.a.bh.1.1 1 180.59 even 6
3600.2.f.g.2449.1 2 180.67 even 12
3600.2.f.g.2449.2 2 180.103 even 12
3600.2.f.q.2449.1 2 180.167 odd 12
3600.2.f.q.2449.2 2 180.23 odd 12