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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [324,2,Mod(109,324)] 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("324.109"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(324, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 324 = 2^{2} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 324.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,0,0,0,-5] 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: \(2.58715302549\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 108)
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label 109.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 324.109
Dual form 324.2.e.b.217.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-2.50000 - 4.33013i) q^{7} +(3.50000 - 6.06218i) q^{13} -1.00000 q^{19} +(2.50000 + 4.33013i) q^{25} +(2.00000 - 3.46410i) q^{31} -1.00000 q^{37} +(-4.00000 - 6.92820i) q^{43} +(-9.00000 + 15.5885i) q^{49} +(6.50000 + 11.2583i) q^{61} +(-5.50000 + 9.52628i) q^{67} +17.0000 q^{73} +(6.50000 + 11.2583i) q^{79} -35.0000 q^{91} +(-2.50000 - 4.33013i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 5 q^{7} + 7 q^{13} - 2 q^{19} + 5 q^{25} + 4 q^{31} - 2 q^{37} - 8 q^{43} - 18 q^{49} + 13 q^{61} - 11 q^{67} + 34 q^{73} + 13 q^{79} - 70 q^{91} - 5 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/324\mathbb{Z}\right)^\times\).

\(n\) \(163\) \(245\)
\(\chi(n)\) \(1\) \(e\left(\frac{1}{3}\right)\)

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 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) −2.50000 − 4.33013i −0.944911 − 1.63663i −0.755929 − 0.654654i \(-0.772814\pi\)
−0.188982 − 0.981981i \(-0.560519\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(12\) 0 0
\(13\) 3.50000 − 6.06218i 0.970725 − 1.68135i 0.277350 − 0.960769i \(-0.410544\pi\)
0.693375 − 0.720577i \(-0.256123\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 − 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(24\) 0 0
\(25\) 2.50000 + 4.33013i 0.500000 + 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(30\) 0 0
\(31\) 2.00000 − 3.46410i 0.359211 − 0.622171i −0.628619 − 0.777714i \(-0.716379\pi\)
0.987829 + 0.155543i \(0.0497126\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −1.00000 −0.164399 −0.0821995 − 0.996616i \(-0.526194\pi\)
−0.0821995 + 0.996616i \(0.526194\pi\)
\(38\) 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 \(-0.957838\pi\)
0.381246 − 0.924473i \(-0.375495\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(48\) 0 0
\(49\) −9.00000 + 15.5885i −1.28571 + 2.22692i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) 6.50000 + 11.2583i 0.832240 + 1.44148i 0.896258 + 0.443533i \(0.146275\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −5.50000 + 9.52628i −0.671932 + 1.16382i 0.305424 + 0.952217i \(0.401202\pi\)
−0.977356 + 0.211604i \(0.932131\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 17.0000 1.98970 0.994850 − 0.101361i \(-0.0323196\pi\)
0.994850 + 0.101361i \(0.0323196\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 6.50000 + 11.2583i 0.731307 + 1.26666i 0.956325 + 0.292306i \(0.0944227\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −35.0000 −3.66900
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −2.50000 − 4.33013i −0.253837 − 0.439658i 0.710742 − 0.703452i \(-0.248359\pi\)
−0.964579 + 0.263795i \(0.915026\pi\)
\(98\) 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 324.2.e.b.109.1 2
3.2 odd 2 CM 324.2.e.b.109.1 2
4.3 odd 2 1296.2.i.j.433.1 2
9.2 odd 6 inner 324.2.e.b.217.1 2
9.4 even 3 108.2.a.a.1.1 ✓ 1
9.5 odd 6 108.2.a.a.1.1 ✓ 1
9.7 even 3 inner 324.2.e.b.217.1 2
12.11 even 2 1296.2.i.j.433.1 2
36.7 odd 6 1296.2.i.j.865.1 2
36.11 even 6 1296.2.i.j.865.1 2
36.23 even 6 432.2.a.d.1.1 1
36.31 odd 6 432.2.a.d.1.1 1
45.4 even 6 2700.2.a.b.1.1 1
45.13 odd 12 2700.2.d.g.649.1 2
45.14 odd 6 2700.2.a.b.1.1 1
45.22 odd 12 2700.2.d.g.649.2 2
45.23 even 12 2700.2.d.g.649.1 2
45.32 even 12 2700.2.d.g.649.2 2
63.13 odd 6 5292.2.a.j.1.1 1
63.41 even 6 5292.2.a.j.1.1 1
72.5 odd 6 1728.2.a.p.1.1 1
72.13 even 6 1728.2.a.p.1.1 1
72.59 even 6 1728.2.a.m.1.1 1
72.67 odd 6 1728.2.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
108.2.a.a.1.1 ✓ 1 9.4 even 3
108.2.a.a.1.1 ✓ 1 9.5 odd 6
324.2.e.b.109.1 2 1.1 even 1 trivial
324.2.e.b.109.1 2 3.2 odd 2 CM
324.2.e.b.217.1 2 9.2 odd 6 inner
324.2.e.b.217.1 2 9.7 even 3 inner
432.2.a.d.1.1 1 36.23 even 6
432.2.a.d.1.1 1 36.31 odd 6
1296.2.i.j.433.1 2 4.3 odd 2
1296.2.i.j.433.1 2 12.11 even 2
1296.2.i.j.865.1 2 36.7 odd 6
1296.2.i.j.865.1 2 36.11 even 6
1728.2.a.m.1.1 1 72.59 even 6
1728.2.a.m.1.1 1 72.67 odd 6
1728.2.a.p.1.1 1 72.5 odd 6
1728.2.a.p.1.1 1 72.13 even 6
2700.2.a.b.1.1 1 45.4 even 6
2700.2.a.b.1.1 1 45.14 odd 6
2700.2.d.g.649.1 2 45.13 odd 12
2700.2.d.g.649.1 2 45.23 even 12
2700.2.d.g.649.2 2 45.22 odd 12
2700.2.d.g.649.2 2 45.32 even 12
5292.2.a.j.1.1 1 63.13 odd 6
5292.2.a.j.1.1 1 63.41 even 6