Properties

Label 11.3.b.a.10.1
Level $11$
Weight $3$
Character 11.10
Self dual yes
Analytic conductor $0.300$
Analytic rank $0$
Dimension $1$
CM discriminant -11
Inner twists $2$

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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] = [11,3,Mod(10,11)] 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("11.10"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(11, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 11.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(0.299728290796\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 10.1
Character \(\chi\) \(=\) 11.10

$q$-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)
 
\(f(q)\) \(=\) \(q-5.00000 q^{3} +4.00000 q^{4} -1.00000 q^{5} +16.0000 q^{9} -11.0000 q^{11} -20.0000 q^{12} +5.00000 q^{15} +16.0000 q^{16} -4.00000 q^{20} +35.0000 q^{23} -24.0000 q^{25} -35.0000 q^{27} -37.0000 q^{31} +55.0000 q^{33} +64.0000 q^{36} -25.0000 q^{37} -44.0000 q^{44} -16.0000 q^{45} +50.0000 q^{47} -80.0000 q^{48} +49.0000 q^{49} -70.0000 q^{53} +11.0000 q^{55} +107.000 q^{59} +20.0000 q^{60} +64.0000 q^{64} +35.0000 q^{67} -175.000 q^{69} -133.000 q^{71} +120.000 q^{75} -16.0000 q^{80} +31.0000 q^{81} -97.0000 q^{89} +140.000 q^{92} +185.000 q^{93} +95.0000 q^{97} -176.000 q^{99} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/11\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) −5.00000 −1.66667 −0.833333 − 0.552771i \(-0.813571\pi\)
−0.833333 + 0.552771i \(0.813571\pi\)
\(4\) 4.00000 1.00000
\(5\) −1.00000 −0.200000 −0.100000 − 0.994987i \(-0.531884\pi\)
−0.100000 + 0.994987i \(0.531884\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 16.0000 1.77778
\(10\) 0 0
\(11\) −11.0000 −1.00000
\(12\) −20.0000 −1.66667
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 5.00000 0.333333
\(16\) 16.0000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −4.00000 −0.200000
\(21\) 0 0
\(22\) 0 0
\(23\) 35.0000 1.52174 0.760870 − 0.648905i \(-0.224773\pi\)
0.760870 + 0.648905i \(0.224773\pi\)
\(24\) 0 0
\(25\) −24.0000 −0.960000
\(26\) 0 0
\(27\) −35.0000 −1.29630
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −37.0000 −1.19355 −0.596774 − 0.802409i \(-0.703551\pi\)
−0.596774 + 0.802409i \(0.703551\pi\)
\(32\) 0 0
\(33\) 55.0000 1.66667
\(34\) 0 0
\(35\) 0 0
\(36\) 64.0000 1.77778
\(37\) −25.0000 −0.675676 −0.337838 − 0.941204i \(-0.609696\pi\)
−0.337838 + 0.941204i \(0.609696\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) −44.0000 −1.00000
\(45\) −16.0000 −0.355556
\(46\) 0 0
\(47\) 50.0000 1.06383 0.531915 − 0.846798i \(-0.321473\pi\)
0.531915 + 0.846798i \(0.321473\pi\)
\(48\) −80.0000 −1.66667
\(49\) 49.0000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −70.0000 −1.32075 −0.660377 − 0.750934i \(-0.729604\pi\)
−0.660377 + 0.750934i \(0.729604\pi\)
\(54\) 0 0
\(55\) 11.0000 0.200000
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 107.000 1.81356 0.906780 − 0.421605i \(-0.138533\pi\)
0.906780 + 0.421605i \(0.138533\pi\)
\(60\) 20.0000 0.333333
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 64.0000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 35.0000 0.522388 0.261194 − 0.965286i \(-0.415884\pi\)
0.261194 + 0.965286i \(0.415884\pi\)
\(68\) 0 0
\(69\) −175.000 −2.53623
\(70\) 0 0
\(71\) −133.000 −1.87324 −0.936620 − 0.350348i \(-0.886063\pi\)
−0.936620 + 0.350348i \(0.886063\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 120.000 1.60000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −16.0000 −0.200000
\(81\) 31.0000 0.382716
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −97.0000 −1.08989 −0.544944 − 0.838473i \(-0.683449\pi\)
−0.544944 + 0.838473i \(0.683449\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 140.000 1.52174
\(93\) 185.000 1.98925
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 95.0000 0.979381 0.489691 − 0.871896i \(-0.337110\pi\)
0.489691 + 0.871896i \(0.337110\pi\)
\(98\) 0 0
\(99\) −176.000 −1.77778
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 11.3.b.a.10.1 ✓ 1
3.2 odd 2 99.3.c.a.10.1 1
4.3 odd 2 176.3.h.a.65.1 1
5.2 odd 4 275.3.d.a.274.2 2
5.3 odd 4 275.3.d.a.274.1 2
5.4 even 2 275.3.c.a.76.1 1
7.6 odd 2 539.3.c.a.197.1 1
8.3 odd 2 704.3.h.a.65.1 1
8.5 even 2 704.3.h.b.65.1 1
11.2 odd 10 121.3.d.b.40.1 4
11.3 even 5 121.3.d.b.112.1 4
11.4 even 5 121.3.d.b.94.1 4
11.5 even 5 121.3.d.b.118.1 4
11.6 odd 10 121.3.d.b.118.1 4
11.7 odd 10 121.3.d.b.94.1 4
11.8 odd 10 121.3.d.b.112.1 4
11.9 even 5 121.3.d.b.40.1 4
11.10 odd 2 CM 11.3.b.a.10.1 ✓ 1
12.11 even 2 1584.3.j.a.1297.1 1
33.32 even 2 99.3.c.a.10.1 1
44.43 even 2 176.3.h.a.65.1 1
55.32 even 4 275.3.d.a.274.2 2
55.43 even 4 275.3.d.a.274.1 2
55.54 odd 2 275.3.c.a.76.1 1
77.76 even 2 539.3.c.a.197.1 1
88.21 odd 2 704.3.h.b.65.1 1
88.43 even 2 704.3.h.a.65.1 1
132.131 odd 2 1584.3.j.a.1297.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
11.3.b.a.10.1 ✓ 1 1.1 even 1 trivial
11.3.b.a.10.1 ✓ 1 11.10 odd 2 CM
99.3.c.a.10.1 1 3.2 odd 2
99.3.c.a.10.1 1 33.32 even 2
121.3.d.b.40.1 4 11.2 odd 10
121.3.d.b.40.1 4 11.9 even 5
121.3.d.b.94.1 4 11.4 even 5
121.3.d.b.94.1 4 11.7 odd 10
121.3.d.b.112.1 4 11.3 even 5
121.3.d.b.112.1 4 11.8 odd 10
121.3.d.b.118.1 4 11.5 even 5
121.3.d.b.118.1 4 11.6 odd 10
176.3.h.a.65.1 1 4.3 odd 2
176.3.h.a.65.1 1 44.43 even 2
275.3.c.a.76.1 1 5.4 even 2
275.3.c.a.76.1 1 55.54 odd 2
275.3.d.a.274.1 2 5.3 odd 4
275.3.d.a.274.1 2 55.43 even 4
275.3.d.a.274.2 2 5.2 odd 4
275.3.d.a.274.2 2 55.32 even 4
539.3.c.a.197.1 1 7.6 odd 2
539.3.c.a.197.1 1 77.76 even 2
704.3.h.a.65.1 1 8.3 odd 2
704.3.h.a.65.1 1 88.43 even 2
704.3.h.b.65.1 1 8.5 even 2
704.3.h.b.65.1 1 88.21 odd 2
1584.3.j.a.1297.1 1 12.11 even 2
1584.3.j.a.1297.1 1 132.131 odd 2