Properties

Label 245.4.a.h
Level 245245
Weight 44
Character orbit 245.a
Self dual yes
Analytic conductor 14.45514.455
Analytic rank 11
Dimension 22
CM no
Inner twists 11

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [245,4,Mod(1,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 245=572 245 = 5 \cdot 7^{2}
Weight: k k == 4 4
Character orbit: [χ][\chi] == 245.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 14.455467951414.4554679514
Analytic rank: 11
Dimension: 22
Coefficient field: Q(2)\Q(\sqrt{2})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x22 x^{2} - 2 Copy content Toggle raw display
Coefficient ring: Z[a1,a2]\Z[a_1, a_2]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 35)
Fricke sign: 1-1
Sato-Tate group: SU(2)\mathrm{SU}(2)

qq-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the qq-expansion are expressed in terms of β=2\beta = \sqrt{2}. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(β3)q2+(3β+1)q3+(6β+3)q45q5+(8β+3)q6+(13β+3)q8+(6β8)q9+(5β+15)q10+(10β+14)q11++(4β+8)q99+O(q100) q + (\beta - 3) q^{2} + (3 \beta + 1) q^{3} + ( - 6 \beta + 3) q^{4} - 5 q^{5} + ( - 8 \beta + 3) q^{6} + (13 \beta + 3) q^{8} + (6 \beta - 8) q^{9} + ( - 5 \beta + 15) q^{10} + (10 \beta + 14) q^{11}+ \cdots + (4 \beta + 8) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 2q6q2+2q3+6q410q5+6q6+6q816q9+30q10+28q1166q12+36q1310q1514q16+76q17+72q18160q1930q2044q22++16q99+O(q100) 2 q - 6 q^{2} + 2 q^{3} + 6 q^{4} - 10 q^{5} + 6 q^{6} + 6 q^{8} - 16 q^{9} + 30 q^{10} + 28 q^{11} - 66 q^{12} + 36 q^{13} - 10 q^{15} - 14 q^{16} + 76 q^{17} + 72 q^{18} - 160 q^{19} - 30 q^{20} - 44 q^{22}+ \cdots + 16 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.

comment: embeddings in the coefficient field
 
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
−1.41421
1.41421
−4.41421 −3.24264 11.4853 −5.00000 14.3137 0 −15.3848 −16.4853 22.0711
1.2 −1.58579 5.24264 −5.48528 −5.00000 −8.31371 0 21.3848 0.485281 7.92893
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
55 +1 +1
77 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 245.4.a.h 2
3.b odd 2 1 2205.4.a.bg 2
5.b even 2 1 1225.4.a.v 2
7.b odd 2 1 245.4.a.g 2
7.c even 3 2 245.4.e.l 4
7.d odd 6 2 35.4.e.b 4
21.c even 2 1 2205.4.a.bf 2
21.g even 6 2 315.4.j.c 4
28.f even 6 2 560.4.q.i 4
35.c odd 2 1 1225.4.a.x 2
35.i odd 6 2 175.4.e.c 4
35.k even 12 4 175.4.k.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.4.e.b 4 7.d odd 6 2
175.4.e.c 4 35.i odd 6 2
175.4.k.c 8 35.k even 12 4
245.4.a.g 2 7.b odd 2 1
245.4.a.h 2 1.a even 1 1 trivial
245.4.e.l 4 7.c even 3 2
315.4.j.c 4 21.g even 6 2
560.4.q.i 4 28.f even 6 2
1225.4.a.v 2 5.b even 2 1
1225.4.a.x 2 35.c odd 2 1
2205.4.a.bf 2 21.c even 2 1
2205.4.a.bg 2 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on S4new(Γ0(245))S_{4}^{\mathrm{new}}(\Gamma_0(245)):

T22+6T2+7 T_{2}^{2} + 6T_{2} + 7 Copy content Toggle raw display
T322T317 T_{3}^{2} - 2T_{3} - 17 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T2+6T+7 T^{2} + 6T + 7 Copy content Toggle raw display
33 T22T17 T^{2} - 2T - 17 Copy content Toggle raw display
55 (T+5)2 (T + 5)^{2} Copy content Toggle raw display
77 T2 T^{2} Copy content Toggle raw display
1111 T228T4 T^{2} - 28T - 4 Copy content Toggle raw display
1313 T236T+124 T^{2} - 36T + 124 Copy content Toggle raw display
1717 T276T4388 T^{2} - 76T - 4388 Copy content Toggle raw display
1919 T2+160T+5048 T^{2} + 160T + 5048 Copy content Toggle raw display
2323 T2+22T27257 T^{2} + 22T - 27257 Copy content Toggle raw display
2929 T2+250T+8425 T^{2} + 250T + 8425 Copy content Toggle raw display
3131 T2132T15644 T^{2} - 132T - 15644 Copy content Toggle raw display
3737 T2+416T+6272 T^{2} + 416T + 6272 Copy content Toggle raw display
4141 T2106T+1009 T^{2} - 106T + 1009 Copy content Toggle raw display
4343 T2+666T+110839 T^{2} + 666T + 110839 Copy content Toggle raw display
4747 T2196T+1412 T^{2} - 196T + 1412 Copy content Toggle raw display
5353 T2+952T+186248 T^{2} + 952T + 186248 Copy content Toggle raw display
5959 T2+840T+34888 T^{2} + 840T + 34888 Copy content Toggle raw display
6161 T2+98T647399 T^{2} + 98T - 647399 Copy content Toggle raw display
6767 T2+1286T+403927 T^{2} + 1286 T + 403927 Copy content Toggle raw display
7171 T21064T+38024 T^{2} - 1064T + 38024 Copy content Toggle raw display
7373 T2172T20452 T^{2} - 172T - 20452 Copy content Toggle raw display
7979 T2+1240T+292808 T^{2} + 1240 T + 292808 Copy content Toggle raw display
8383 T21906T+908159 T^{2} - 1906 T + 908159 Copy content Toggle raw display
8989 T2+650T104327 T^{2} + 650T - 104327 Copy content Toggle raw display
9797 T2628T401404 T^{2} - 628T - 401404 Copy content Toggle raw display
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