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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [189,2,Mod(26,189)] 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("189.26"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(189, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 5])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 189 = 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 189.p (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.50917259820\)
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: yes
Sato-Tate group: $\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label 26.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 189.26
Dual form 189.2.p.a.80.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 1.73205i) q^{4} +(0.500000 - 2.59808i) q^{7} -6.92820i q^{13} +(-2.00000 + 3.46410i) q^{16} +(4.50000 + 2.59808i) q^{19} +(2.50000 + 4.33013i) q^{25} +(-5.00000 + 1.73205i) q^{28} +(-1.50000 + 0.866025i) q^{31} +(-5.00000 + 8.66025i) q^{37} +13.0000 q^{43} +(-6.50000 - 2.59808i) q^{49} +(-12.0000 + 6.92820i) q^{52} +(-7.50000 - 4.33013i) q^{61} +8.00000 q^{64} +(8.00000 + 13.8564i) q^{67} +(13.5000 - 7.79423i) q^{73} -10.3923i q^{76} +(2.00000 - 3.46410i) q^{79} +(-18.0000 - 3.46410i) q^{91} -19.0526i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{4} + q^{7} - 4 q^{16} + 9 q^{19} + 5 q^{25} - 10 q^{28} - 3 q^{31} - 10 q^{37} + 26 q^{43} - 13 q^{49} - 24 q^{52} - 15 q^{61} + 16 q^{64} + 16 q^{67} + 27 q^{73} + 4 q^{79} - 36 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(136\)
\(\chi(n)\) \(-1\) \(e\left(\frac{5}{6}\right)\)

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 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(3\) 0 0
\(4\) −1.00000 − 1.73205i −0.500000 − 0.866025i
\(5\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(6\) 0 0
\(7\) 0.500000 − 2.59808i 0.188982 − 0.981981i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(12\) 0 0
\(13\) − 6.92820i − 1.92154i −0.277350 − 0.960769i \(-0.589456\pi\)
0.277350 − 0.960769i \(-0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −2.00000 + 3.46410i −0.500000 + 0.866025i
\(17\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(18\) 0 0
\(19\) 4.50000 + 2.59808i 1.03237 + 0.596040i 0.917663 − 0.397360i \(-0.130073\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.500000 − 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) 2.50000 + 4.33013i 0.500000 + 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) −5.00000 + 1.73205i −0.944911 + 0.327327i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −1.50000 + 0.866025i −0.269408 + 0.155543i −0.628619 − 0.777714i \(-0.716379\pi\)
0.359211 + 0.933257i \(0.383046\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.00000 + 8.66025i −0.821995 + 1.42374i 0.0821995 + 0.996616i \(0.473806\pi\)
−0.904194 + 0.427121i \(0.859528\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 13.0000 1.98248 0.991241 − 0.132068i \(-0.0421616\pi\)
0.991241 + 0.132068i \(0.0421616\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(48\) 0 0
\(49\) −6.50000 − 2.59808i −0.928571 − 0.371154i
\(50\) 0 0
\(51\) 0 0
\(52\) −12.0000 + 6.92820i −1.66410 + 0.960769i
\(53\) 0 0 −0.500000 − 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(60\) 0 0
\(61\) −7.50000 − 4.33013i −0.960277 − 0.554416i −0.0640184 − 0.997949i \(-0.520392\pi\)
−0.896258 + 0.443533i \(0.853725\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 8.00000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 8.00000 + 13.8564i 0.977356 + 1.69283i 0.671932 + 0.740613i \(0.265465\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 13.5000 − 7.79423i 1.58006 − 0.912245i 0.585206 − 0.810885i \(-0.301014\pi\)
0.994850 − 0.101361i \(-0.0323196\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) − 10.3923i − 1.19208i
\(77\) 0 0
\(78\) 0 0
\(79\) 2.00000 − 3.46410i 0.225018 − 0.389742i −0.731307 − 0.682048i \(-0.761089\pi\)
0.956325 + 0.292306i \(0.0944227\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(90\) 0 0
\(91\) −18.0000 − 3.46410i −1.88691 − 0.363137i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 19.0526i − 1.93449i −0.253837 − 0.967247i \(-0.581693\pi\)
0.253837 − 0.967247i \(-0.418307\pi\)
\(98\) 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 189.2.p.a.26.1 ✓ 2
3.2 odd 2 CM 189.2.p.a.26.1 ✓ 2
7.2 even 3 1323.2.c.a.1322.1 2
7.3 odd 6 inner 189.2.p.a.80.1 yes 2
7.5 odd 6 1323.2.c.a.1322.2 2
9.2 odd 6 567.2.i.a.215.1 2
9.4 even 3 567.2.s.b.26.1 2
9.5 odd 6 567.2.s.b.26.1 2
9.7 even 3 567.2.i.a.215.1 2
21.2 odd 6 1323.2.c.a.1322.1 2
21.5 even 6 1323.2.c.a.1322.2 2
21.17 even 6 inner 189.2.p.a.80.1 yes 2
63.31 odd 6 567.2.i.a.269.1 2
63.38 even 6 567.2.s.b.458.1 2
63.52 odd 6 567.2.s.b.458.1 2
63.59 even 6 567.2.i.a.269.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
189.2.p.a.26.1 ✓ 2 1.1 even 1 trivial
189.2.p.a.26.1 ✓ 2 3.2 odd 2 CM
189.2.p.a.80.1 yes 2 7.3 odd 6 inner
189.2.p.a.80.1 yes 2 21.17 even 6 inner
567.2.i.a.215.1 2 9.2 odd 6
567.2.i.a.215.1 2 9.7 even 3
567.2.i.a.269.1 2 63.31 odd 6
567.2.i.a.269.1 2 63.59 even 6
567.2.s.b.26.1 2 9.4 even 3
567.2.s.b.26.1 2 9.5 odd 6
567.2.s.b.458.1 2 63.38 even 6
567.2.s.b.458.1 2 63.52 odd 6
1323.2.c.a.1322.1 2 7.2 even 3
1323.2.c.a.1322.1 2 21.2 odd 6
1323.2.c.a.1322.2 2 7.5 odd 6
1323.2.c.a.1322.2 2 21.5 even 6