Properties

Label 11.3.d.a
Level 1111
Weight 33
Character orbit 11.d
Analytic conductor 0.3000.300
Analytic rank 00
Dimension 44
Inner twists 22

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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(2,11)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(11, base_ring=CyclotomicField(10)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 3, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("11.2"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Level: N N == 11 11
Weight: k k == 3 3
Character orbit: [χ][\chi] == 11.d (of order 1010, degree 44, 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: no
Analytic conductor: 0.2997282907960.299728290796
Analytic rank: 00
Dimension: 44
Coefficient field: Q(ζ10)\Q(\zeta_{10})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x4x3+x2x+1 x^{4} - x^{3} + x^{2} - x + 1 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 1 1
Twist minimal: yes
Sato-Tate group: SU(2)[C10]\mathrm{SU}(2)[C_{10}]

qq-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)
 

Coefficients of the qq-expansion are expressed in terms of a primitive root of unity ζ10\zeta_{10}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(ζ103+2ζ10)q2+(2ζ1033ζ102+2)q3+(4ζ102+3ζ104)q4+4ζ102q5+(ζ103+3ζ102++6)q6++(69ζ103+7ζ102++28)q99+O(q100) q + ( - \zeta_{10}^{3} + \cdots - 2 \zeta_{10}) q^{2} + (2 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + \cdots - 2) q^{3} + ( - 4 \zeta_{10}^{2} + 3 \zeta_{10} - 4) q^{4} + 4 \zeta_{10}^{2} q^{5} + (\zeta_{10}^{3} + 3 \zeta_{10}^{2} + \cdots + 6) q^{6}+ \cdots + ( - 69 \zeta_{10}^{3} + 7 \zeta_{10}^{2} + \cdots + 28) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q5q29q44q5+15q6+10q7+15q811q9+q1130q1220q1310q14+19q16+30q18+25q19+44q2035q2220q23+5q24++31q99+O(q100) 4 q - 5 q^{2} - 9 q^{4} - 4 q^{5} + 15 q^{6} + 10 q^{7} + 15 q^{8} - 11 q^{9} + q^{11} - 30 q^{12} - 20 q^{13} - 10 q^{14} + 19 q^{16} + 30 q^{18} + 25 q^{19} + 44 q^{20} - 35 q^{22} - 20 q^{23} + 5 q^{24}+ \cdots + 31 q^{99}+O(q^{100}) Copy content Toggle raw display

Character values

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

nn 22
χ(n)\chi(n) ζ10\zeta_{10}

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\iota_m(a_n) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
2.1
0.809017 + 0.587785i
0.809017 0.587785i
−0.309017 0.951057i
−0.309017 + 0.951057i
−0.690983 0.224514i −1.11803 + 0.812299i −2.80902 2.04087i 1.23607 + 3.80423i 0.954915 0.310271i 5.85410 8.05748i 3.19098 + 4.39201i −2.19098 + 6.74315i 2.90617i
6.1 −0.690983 + 0.224514i −1.11803 0.812299i −2.80902 + 2.04087i 1.23607 3.80423i 0.954915 + 0.310271i 5.85410 + 8.05748i 3.19098 4.39201i −2.19098 6.74315i 2.90617i
7.1 −1.80902 + 2.48990i 1.11803 3.44095i −1.69098 5.20431i −3.23607 + 2.35114i 6.54508 + 9.00854i −0.854102 + 0.277515i 4.30902 + 1.40008i −3.30902 2.40414i 12.3107i
8.1 −1.80902 2.48990i 1.11803 + 3.44095i −1.69098 + 5.20431i −3.23607 2.35114i 6.54508 9.00854i −0.854102 0.277515i 4.30902 1.40008i −3.30902 + 2.40414i 12.3107i
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.d odd 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 11.3.d.a 4
3.b odd 2 1 99.3.k.a 4
4.b odd 2 1 176.3.n.a 4
5.b even 2 1 275.3.x.e 4
5.c odd 4 2 275.3.q.d 8
11.b odd 2 1 121.3.d.d 4
11.c even 5 1 121.3.b.b 4
11.c even 5 1 121.3.d.a 4
11.c even 5 1 121.3.d.c 4
11.c even 5 1 121.3.d.d 4
11.d odd 10 1 inner 11.3.d.a 4
11.d odd 10 1 121.3.b.b 4
11.d odd 10 1 121.3.d.a 4
11.d odd 10 1 121.3.d.c 4
33.f even 10 1 99.3.k.a 4
33.f even 10 1 1089.3.c.e 4
33.h odd 10 1 1089.3.c.e 4
44.g even 10 1 176.3.n.a 4
55.h odd 10 1 275.3.x.e 4
55.l even 20 2 275.3.q.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.3.d.a 4 1.a even 1 1 trivial
11.3.d.a 4 11.d odd 10 1 inner
99.3.k.a 4 3.b odd 2 1
99.3.k.a 4 33.f even 10 1
121.3.b.b 4 11.c even 5 1
121.3.b.b 4 11.d odd 10 1
121.3.d.a 4 11.c even 5 1
121.3.d.a 4 11.d odd 10 1
121.3.d.c 4 11.c even 5 1
121.3.d.c 4 11.d odd 10 1
121.3.d.d 4 11.b odd 2 1
121.3.d.d 4 11.c even 5 1
176.3.n.a 4 4.b odd 2 1
176.3.n.a 4 44.g even 10 1
275.3.q.d 8 5.c odd 4 2
275.3.q.d 8 55.l even 20 2
275.3.x.e 4 5.b even 2 1
275.3.x.e 4 55.h odd 10 1
1089.3.c.e 4 33.f even 10 1
1089.3.c.e 4 33.h odd 10 1

Hecke kernels

This newform subspace is the entire newspace S3new(11,[χ])S_{3}^{\mathrm{new}}(11, [\chi]).

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T4+5T3++5 T^{4} + 5 T^{3} + \cdots + 5 Copy content Toggle raw display
33 T4+10T2++25 T^{4} + 10 T^{2} + \cdots + 25 Copy content Toggle raw display
55 T4+4T3++256 T^{4} + 4 T^{3} + \cdots + 256 Copy content Toggle raw display
77 T410T3++80 T^{4} - 10 T^{3} + \cdots + 80 Copy content Toggle raw display
1111 T4T3++14641 T^{4} - T^{3} + \cdots + 14641 Copy content Toggle raw display
1313 T4+20T3++2000 T^{4} + 20 T^{3} + \cdots + 2000 Copy content Toggle raw display
1717 T410985T+142805 T^{4} - 10985 T + 142805 Copy content Toggle raw display
1919 T425T3++605 T^{4} - 25 T^{3} + \cdots + 605 Copy content Toggle raw display
2323 (T2+10T+20)2 (T^{2} + 10 T + 20)^{2} Copy content Toggle raw display
2929 T4+40T3++9680 T^{4} + 40 T^{3} + \cdots + 9680 Copy content Toggle raw display
3131 T4+58T3++55696 T^{4} + 58 T^{3} + \cdots + 55696 Copy content Toggle raw display
3737 T490T3++2624400 T^{4} - 90 T^{3} + \cdots + 2624400 Copy content Toggle raw display
4141 T4+80T3++8405 T^{4} + 80 T^{3} + \cdots + 8405 Copy content Toggle raw display
4343 T4+1625T2+581405 T^{4} + 1625 T^{2} + 581405 Copy content Toggle raw display
4747 T4+30T3++384400 T^{4} + 30 T^{3} + \cdots + 384400 Copy content Toggle raw display
5353 T4120T3++810000 T^{4} - 120 T^{3} + \cdots + 810000 Copy content Toggle raw display
5959 T423T3++5041 T^{4} - 23 T^{3} + \cdots + 5041 Copy content Toggle raw display
6161 T410T3++403280 T^{4} - 10 T^{3} + \cdots + 403280 Copy content Toggle raw display
6767 (T2+115T+2945)2 (T^{2} + 115 T + 2945)^{2} Copy content Toggle raw display
7171 T4148T3++22619536 T^{4} - 148 T^{3} + \cdots + 22619536 Copy content Toggle raw display
7373 T4300T3++93787805 T^{4} - 300 T^{3} + \cdots + 93787805 Copy content Toggle raw display
7979 T470T3++67280 T^{4} - 70 T^{3} + \cdots + 67280 Copy content Toggle raw display
8383 T4225T3++22281605 T^{4} - 225 T^{3} + \cdots + 22281605 Copy content Toggle raw display
8989 (T261T7681)2 (T^{2} - 61 T - 7681)^{2} Copy content Toggle raw display
9797 T4+165T3++31416025 T^{4} + 165 T^{3} + \cdots + 31416025 Copy content Toggle raw display
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