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Properties

Label 441.2.e.c.361.1
Level $441$
Weight $2$
Character 441.361
Analytic conductor $3.521$
Analytic rank $0$
Dimension $2$
CM discriminant -3
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] = [441,2,Mod(226,441)] 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("441.226"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(441, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 2])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 441.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,0,2,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(10)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(3.52140272914\)
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: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

Embedding invariants

Embedding label 361.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 441.361
Dual form 441.2.e.c.226.1

$q$-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)
 
\(f(q)\) \(=\) \(q+(1.00000 - 1.73205i) q^{4} +7.00000 q^{13} +(-2.00000 - 3.46410i) q^{16} +(-3.50000 - 6.06218i) q^{19} +(2.50000 - 4.33013i) q^{25} +(-3.50000 + 6.06218i) q^{31} +(0.500000 + 0.866025i) q^{37} +5.00000 q^{43} +(7.00000 - 12.1244i) q^{52} +(7.00000 + 12.1244i) q^{61} -8.00000 q^{64} +(-5.50000 + 9.52628i) q^{67} +(-3.50000 + 6.06218i) q^{73} -14.0000 q^{76} +(6.50000 + 11.2583i) q^{79} -14.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 14 q^{13} - 4 q^{16} - 7 q^{19} + 5 q^{25} - 7 q^{31} + q^{37} + 10 q^{43} + 14 q^{52} + 14 q^{61} - 16 q^{64} - 11 q^{67} - 7 q^{73} - 28 q^{76} + 13 q^{79} - 28 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/441\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(344\)
\(\chi(n)\) \(e\left(\frac{2}{3}\right)\) \(1\)

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 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(3\) 0 0
\(4\) 1.00000 − 1.73205i 0.500000 − 0.866025i
\(5\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(12\) 0 0
\(13\) 7.00000 1.94145 0.970725 − 0.240192i \(-0.0772105\pi\)
0.970725 + 0.240192i \(0.0772105\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −2.00000 − 3.46410i −0.500000 − 0.866025i
\(17\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(18\) 0 0
\(19\) −3.50000 − 6.06218i −0.802955 − 1.39076i −0.917663 − 0.397360i \(-0.869927\pi\)
0.114708 − 0.993399i \(-0.463407\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.866025 − 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(24\) 0 0
\(25\) 2.50000 − 4.33013i 0.500000 − 0.866025i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −3.50000 + 6.06218i −0.628619 + 1.08880i 0.359211 + 0.933257i \(0.383046\pi\)
−0.987829 + 0.155543i \(0.950287\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.500000 + 0.866025i 0.0821995 + 0.142374i 0.904194 − 0.427121i \(-0.140472\pi\)
−0.821995 + 0.569495i \(0.807139\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 5.00000 0.762493 0.381246 − 0.924473i \(-0.375495\pi\)
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
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 7.00000 − 12.1244i 0.970725 − 1.68135i
\(53\) 0 0 −0.866025 − 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(54\) 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
\(61\) 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
\(64\) −8.00000 −1.00000
\(65\) 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
\(73\) −3.50000 + 6.06218i −0.409644 + 0.709524i −0.994850 − 0.101361i \(-0.967680\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −14.0000 −1.60591
\(77\) 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 − 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 \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −14.0000 −1.42148 −0.710742 − 0.703452i \(-0.751641\pi\)
−0.710742 + 0.703452i \(0.751641\pi\)
\(98\) 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 441.2.e.c.361.1 2
3.2 odd 2 CM 441.2.e.c.361.1 2
7.2 even 3 inner 441.2.e.c.226.1 2
7.3 odd 6 441.2.a.d.1.1 1
7.4 even 3 441.2.a.e.1.1 1
7.5 odd 6 63.2.e.a.37.1 ✓ 2
7.6 odd 2 63.2.e.a.46.1 yes 2
21.2 odd 6 inner 441.2.e.c.226.1 2
21.5 even 6 63.2.e.a.37.1 ✓ 2
21.11 odd 6 441.2.a.e.1.1 1
21.17 even 6 441.2.a.d.1.1 1
21.20 even 2 63.2.e.a.46.1 yes 2
28.3 even 6 7056.2.a.y.1.1 1
28.11 odd 6 7056.2.a.bf.1.1 1
28.19 even 6 1008.2.s.j.289.1 2
28.27 even 2 1008.2.s.j.865.1 2
63.5 even 6 567.2.g.d.541.1 2
63.13 odd 6 567.2.h.c.298.1 2
63.20 even 6 567.2.g.d.109.1 2
63.34 odd 6 567.2.g.d.109.1 2
63.40 odd 6 567.2.g.d.541.1 2
63.41 even 6 567.2.h.c.298.1 2
63.47 even 6 567.2.h.c.352.1 2
63.61 odd 6 567.2.h.c.352.1 2
84.11 even 6 7056.2.a.bf.1.1 1
84.47 odd 6 1008.2.s.j.289.1 2
84.59 odd 6 7056.2.a.y.1.1 1
84.83 odd 2 1008.2.s.j.865.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.2.e.a.37.1 ✓ 2 7.5 odd 6
63.2.e.a.37.1 ✓ 2 21.5 even 6
63.2.e.a.46.1 yes 2 7.6 odd 2
63.2.e.a.46.1 yes 2 21.20 even 2
441.2.a.d.1.1 1 7.3 odd 6
441.2.a.d.1.1 1 21.17 even 6
441.2.a.e.1.1 1 7.4 even 3
441.2.a.e.1.1 1 21.11 odd 6
441.2.e.c.226.1 2 7.2 even 3 inner
441.2.e.c.226.1 2 21.2 odd 6 inner
441.2.e.c.361.1 2 1.1 even 1 trivial
441.2.e.c.361.1 2 3.2 odd 2 CM
567.2.g.d.109.1 2 63.20 even 6
567.2.g.d.109.1 2 63.34 odd 6
567.2.g.d.541.1 2 63.5 even 6
567.2.g.d.541.1 2 63.40 odd 6
567.2.h.c.298.1 2 63.13 odd 6
567.2.h.c.298.1 2 63.41 even 6
567.2.h.c.352.1 2 63.47 even 6
567.2.h.c.352.1 2 63.61 odd 6
1008.2.s.j.289.1 2 28.19 even 6
1008.2.s.j.289.1 2 84.47 odd 6
1008.2.s.j.865.1 2 28.27 even 2
1008.2.s.j.865.1 2 84.83 odd 2
7056.2.a.y.1.1 1 28.3 even 6
7056.2.a.y.1.1 1 84.59 odd 6
7056.2.a.bf.1.1 1 28.11 odd 6
7056.2.a.bf.1.1 1 84.11 even 6