Properties

Label 50.7.c.e
Level 5050
Weight 77
Character orbit 50.c
Analytic conductor 11.50311.503
Analytic rank 00
Dimension 44
Inner twists 22

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [50,7,Mod(7,50)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(50, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("50.7");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: N N == 50=252 50 = 2 \cdot 5^{2}
Weight: k k == 7 7
Character orbit: [χ][\chi] == 50.c (of order 44, degree 22, minimal)

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: no
Analytic conductor: 11.502704181011.5027041810
Analytic rank: 00
Dimension: 44
Relative dimension: 22 over Q(i)\Q(i)
Coefficient field: Q(i,6)\Q(i, \sqrt{6})
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4+9 x^{4} + 9 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 52 5^{2}
Twist minimal: yes
Sato-Tate group: SU(2)[C4]\mathrm{SU}(2)[C_{4}]

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 a basis 1,β1,β2,β31,\beta_1,\beta_2,\beta_3 for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(4β24)q2+(β312β2+12)q3+32β2q4+(4β34β196)q6+(168β2+44β1+168)q7+(128β2+128)q8++(55440β3631458β2+55440β1)q99+O(q100) q + ( - 4 \beta_{2} - 4) q^{2} + ( - \beta_{3} - 12 \beta_{2} + 12) q^{3} + 32 \beta_{2} q^{4} + (4 \beta_{3} - 4 \beta_1 - 96) q^{6} + (168 \beta_{2} + 44 \beta_1 + 168) q^{7} + ( - 128 \beta_{2} + 128) q^{8}+ \cdots + (55440 \beta_{3} - 631458 \beta_{2} + 55440 \beta_1) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q16q2+48q3384q6+672q7+512q82652q11+1536q121632q134096q16+18912q17+5856q18+29328q21+10608q22+19488q23+13056q26++1343984q98+O(q100) 4 q - 16 q^{2} + 48 q^{3} - 384 q^{6} + 672 q^{7} + 512 q^{8} - 2652 q^{11} + 1536 q^{12} - 1632 q^{13} - 4096 q^{16} + 18912 q^{17} + 5856 q^{18} + 29328 q^{21} + 10608 q^{22} + 19488 q^{23} + 13056 q^{26}+ \cdots + 1343984 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x4+9 x^{4} + 9 : Copy content Toggle raw display

β1\beta_{1}== 5ν 5\nu Copy content Toggle raw display
β2\beta_{2}== (ν2)/3 ( \nu^{2} ) / 3 Copy content Toggle raw display
β3\beta_{3}== (5ν3)/3 ( 5\nu^{3} ) / 3 Copy content Toggle raw display
ν\nu== (β1)/5 ( \beta_1 ) / 5 Copy content Toggle raw display
ν2\nu^{2}== 3β2 3\beta_{2} Copy content Toggle raw display
ν3\nu^{3}== (3β3)/5 ( 3\beta_{3} ) / 5 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/50Z)×\left(\mathbb{Z}/50\mathbb{Z}\right)^\times.

nn 2727
χ(n)\chi(n) β2\beta_{2}

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}
7.1
−1.22474 1.22474i
1.22474 + 1.22474i
−1.22474 + 1.22474i
1.22474 1.22474i
−4.00000 4.00000i 5.87628 5.87628i 32.0000i 0 −47.0102 −101.444 101.444i 128.000 128.000i 659.939i 0
7.2 −4.00000 4.00000i 18.1237 18.1237i 32.0000i 0 −144.990 437.444 + 437.444i 128.000 128.000i 72.0612i 0
43.1 −4.00000 + 4.00000i 5.87628 + 5.87628i 32.0000i 0 −47.0102 −101.444 + 101.444i 128.000 + 128.000i 659.939i 0
43.2 −4.00000 + 4.00000i 18.1237 + 18.1237i 32.0000i 0 −144.990 437.444 437.444i 128.000 + 128.000i 72.0612i 0
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 50.7.c.e 4
3.b odd 2 1 450.7.g.n 4
5.b even 2 1 50.7.c.f yes 4
5.c odd 4 1 inner 50.7.c.e 4
5.c odd 4 1 50.7.c.f yes 4
15.d odd 2 1 450.7.g.e 4
15.e even 4 1 450.7.g.e 4
15.e even 4 1 450.7.g.n 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.7.c.e 4 1.a even 1 1 trivial
50.7.c.e 4 5.c odd 4 1 inner
50.7.c.f yes 4 5.b even 2 1
50.7.c.f yes 4 5.c odd 4 1
450.7.g.e 4 15.d odd 2 1
450.7.g.e 4 15.e even 4 1
450.7.g.n 4 3.b odd 2 1
450.7.g.n 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3448T33+1152T3210224T3+45369 T_{3}^{4} - 48T_{3}^{3} + 1152T_{3}^{2} - 10224T_{3} + 45369 acting on S7new(50,[χ])S_{7}^{\mathrm{new}}(50, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 (T2+8T+32)2 (T^{2} + 8 T + 32)^{2} Copy content Toggle raw display
33 T448T3++45369 T^{4} - 48 T^{3} + \cdots + 45369 Copy content Toggle raw display
55 T4 T^{4} Copy content Toggle raw display
77 T4++7876917504 T^{4} + \cdots + 7876917504 Copy content Toggle raw display
1111 (T2+1326T1310031)2 (T^{2} + 1326 T - 1310031)^{2} Copy content Toggle raw display
1313 T4++1199839818384 T^{4} + \cdots + 1199839818384 Copy content Toggle raw display
1717 T4++19 ⁣ ⁣49 T^{4} + \cdots + 19\!\cdots\!49 Copy content Toggle raw display
1919 T4++16 ⁣ ⁣25 T^{4} + \cdots + 16\!\cdots\!25 Copy content Toggle raw display
2323 T4++32 ⁣ ⁣24 T^{4} + \cdots + 32\!\cdots\!24 Copy content Toggle raw display
2929 T4++18 ⁣ ⁣00 T^{4} + \cdots + 18\!\cdots\!00 Copy content Toggle raw display
3131 (T2+14236T+12563524)2 (T^{2} + 14236 T + 12563524)^{2} Copy content Toggle raw display
3737 T4++53 ⁣ ⁣64 T^{4} + \cdots + 53\!\cdots\!64 Copy content Toggle raw display
4141 (T2+43566T3121123311)2 (T^{2} + 43566 T - 3121123311)^{2} Copy content Toggle raw display
4343 T4++13 ⁣ ⁣04 T^{4} + \cdots + 13\!\cdots\!04 Copy content Toggle raw display
4747 T4++60 ⁣ ⁣84 T^{4} + \cdots + 60\!\cdots\!84 Copy content Toggle raw display
5353 T4++75 ⁣ ⁣44 T^{4} + \cdots + 75\!\cdots\!44 Copy content Toggle raw display
5959 T4++18 ⁣ ⁣00 T^{4} + \cdots + 18\!\cdots\!00 Copy content Toggle raw display
6161 (T2100024T9135929456)2 (T^{2} - 100024 T - 9135929456)^{2} Copy content Toggle raw display
6767 T4++21 ⁣ ⁣49 T^{4} + \cdots + 21\!\cdots\!49 Copy content Toggle raw display
7171 (T2+548556T+41153643684)2 (T^{2} + 548556 T + 41153643684)^{2} Copy content Toggle raw display
7373 T4++24 ⁣ ⁣49 T^{4} + \cdots + 24\!\cdots\!49 Copy content Toggle raw display
7979 T4++84 ⁣ ⁣00 T^{4} + \cdots + 84\!\cdots\!00 Copy content Toggle raw display
8383 T4++45 ⁣ ⁣89 T^{4} + \cdots + 45\!\cdots\!89 Copy content Toggle raw display
8989 T4++15 ⁣ ⁣25 T^{4} + \cdots + 15\!\cdots\!25 Copy content Toggle raw display
9797 T4++20 ⁣ ⁣84 T^{4} + \cdots + 20\!\cdots\!84 Copy content Toggle raw display
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