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Properties

Label 275.2.a.e
Level 275275
Weight 22
Character orbit 275.a
Self dual yes
Analytic conductor 2.1962.196
Analytic rank 11
Dimension 22
CM no
Inner twists 11

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [275,2,Mod(1,275)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(275, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("275.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 275=52⋅11 275 = 5^{2} \cdot 11
Weight: k k == 2 2
Character orbit: [χ][\chi] == 275.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,-1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 2.195886055592.19588605559
Analytic rank: 11
Dimension: 22
Coefficient field: Q(13)\Q(\sqrt{13})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x2−x−3 x^{2} - x - 3 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 1 1
Twist minimal: yes
Fricke sign: +1+1
Sato-Tate group: SU(2)\mathrm{SU}(2)

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 β=12(1+13)\beta = \frac{1}{2}(1 + \sqrt{13}). We also show the integral qq-expansion of the trace form.

f(q)f(q) == q−βq2+(β−1)q3+(β+1)q4−3q6+(−β−2)q7−3q8+(−β+1)q9−q11+(β+2)q12−5q13+(3β+3)q14+(β−2)q16+⋯+(β−1)q99+O(q100) q - \beta q^{2} + (\beta - 1) q^{3} + (\beta + 1) q^{4} - 3 q^{6} + ( - \beta - 2) q^{7} - 3 q^{8} + ( - \beta + 1) q^{9} - q^{11} + (\beta + 2) q^{12} - 5 q^{13} + (3 \beta + 3) q^{14} + (\beta - 2) q^{16} + \cdots + (\beta - 1) q^{99} +O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 2q−q2−q3+3q4−6q6−5q7−6q8+q9−2q11+5q12−10q13+9q14−3q16−3q17+6q18−2q19−4q21+q22+11q23+3q24+⋯−q99+O(q100) 2 q - q^{2} - q^{3} + 3 q^{4} - 6 q^{6} - 5 q^{7} - 6 q^{8} + q^{9} - 2 q^{11} + 5 q^{12} - 10 q^{13} + 9 q^{14} - 3 q^{16} - 3 q^{17} + 6 q^{18} - 2 q^{19} - 4 q^{21} + q^{22} + 11 q^{23} + 3 q^{24}+ \cdots - q^{99}+O(q^{100}) Copy content Toggle raw display

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}
1.1
2.30278
−1.30278
−2.30278 1.30278 3.30278 0 −3.00000 −4.30278 −3.00000 −1.30278 0
1.2 1.30278 −2.30278 −0.302776 0 −3.00000 −0.697224 −3.00000 2.30278 0
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
55 +1 +1
1111 +1 +1

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 275.2.a.e ✓ 2
3.b odd 2 1 2475.2.a.t 2
4.b odd 2 1 4400.2.a.bs 2
5.b even 2 1 275.2.a.f yes 2
5.c odd 4 2 275.2.b.c 4
11.b odd 2 1 3025.2.a.n 2
15.d odd 2 1 2475.2.a.o 2
15.e even 4 2 2475.2.c.k 4
20.d odd 2 1 4400.2.a.bh 2
20.e even 4 2 4400.2.b.y 4
55.d odd 2 1 3025.2.a.h 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.a.e ✓ 2 1.a even 1 1 trivial
275.2.a.f yes 2 5.b even 2 1
275.2.b.c 4 5.c odd 4 2
2475.2.a.o 2 15.d odd 2 1
2475.2.a.t 2 3.b odd 2 1
2475.2.c.k 4 15.e even 4 2
3025.2.a.h 2 55.d odd 2 1
3025.2.a.n 2 11.b odd 2 1
4400.2.a.bh 2 20.d odd 2 1
4400.2.a.bs 2 4.b odd 2 1
4400.2.b.y 4 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T22+T2−3 T_{2}^{2} + T_{2} - 3 acting on S2new(Γ0(275))S_{2}^{\mathrm{new}}(\Gamma_0(275)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2+T−3 T^{2} + T - 3 Copy content Toggle raw display
33 T2+T−3 T^{2} + T - 3 Copy content Toggle raw display
55 T2 T^{2} Copy content Toggle raw display
77 T2+5T+3 T^{2} + 5T + 3 Copy content Toggle raw display
1111 (T+1)2 (T + 1)^{2} Copy content Toggle raw display
1313 (T+5)2 (T + 5)^{2} Copy content Toggle raw display
1717 T2+3T−27 T^{2} + 3T - 27 Copy content Toggle raw display
1919 (T+1)2 (T + 1)^{2} Copy content Toggle raw display
2323 T2−11T+27 T^{2} - 11T + 27 Copy content Toggle raw display
2929 T2+9T−9 T^{2} + 9T - 9 Copy content Toggle raw display
3131 T2−6T−43 T^{2} - 6T - 43 Copy content Toggle raw display
3737 T2+12T+23 T^{2} + 12T + 23 Copy content Toggle raw display
4141 T2+4T−9 T^{2} + 4T - 9 Copy content Toggle raw display
4343 T2−52 T^{2} - 52 Copy content Toggle raw display
4747 (T−3)2 (T - 3)^{2} Copy content Toggle raw display
5353 T2+T−3 T^{2} + T - 3 Copy content Toggle raw display
5959 T2+14T−3 T^{2} + 14T - 3 Copy content Toggle raw display
6161 T2+5T−23 T^{2} + 5T - 23 Copy content Toggle raw display
6767 (T−4)2 (T - 4)^{2} Copy content Toggle raw display
7171 T2−2T−12 T^{2} - 2T - 12 Copy content Toggle raw display
7373 T2−5T−23 T^{2} - 5T - 23 Copy content Toggle raw display
7979 T2+11T+1 T^{2} + 11T + 1 Copy content Toggle raw display
8383 T2+11T−51 T^{2} + 11T - 51 Copy content Toggle raw display
8989 T2−7T+9 T^{2} - 7T + 9 Copy content Toggle raw display
9797 T2+27T+179 T^{2} + 27T + 179 Copy content Toggle raw display
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