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Properties

Label 1127.2.a.l
Level 11271127
Weight 22
Character orbit 1127.a
Self dual yes
Analytic conductor 8.9998.999
Analytic rank 11
Dimension 77
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] = [1127,2,Mod(1,1127)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1127, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1127.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 1127=72⋅23 1127 = 7^{2} \cdot 23
Weight: k k == 2 2
Character orbit: [χ][\chi] == 1127.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: 8.999140307808.99914030780
Analytic rank: 11
Dimension: 77
Coefficient field: Q[x]/(x7−⋯ )\mathbb{Q}[x]/(x^{7} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x7−10x5−x4+25x3+x2−8x+1 x^{7} - 10x^{5} - x^{4} + 25x^{3} + x^{2} - 8x + 1 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 161)
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 a basis 1,β1,…,β61,\beta_1,\ldots,\beta_{6} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q−β1q2−β6q3+(β6+β5+β3+⋯+1)q4+(β2−1)q5+(β6+β5+β4+⋯−1)q6+(−β6−β5+⋯−β1)q8+⋯+(2β6+β5−2β3+⋯+5)q99+O(q100) q - \beta_1 q^{2} - \beta_{6} q^{3} + (\beta_{6} + \beta_{5} + \beta_{3} + \cdots + 1) q^{4} + (\beta_{2} - 1) q^{5} + (\beta_{6} + \beta_{5} + \beta_{4} + \cdots - 1) q^{6} + ( - \beta_{6} - \beta_{5} + \cdots - \beta_1) q^{8}+ \cdots + (2 \beta_{6} + \beta_{5} - 2 \beta_{3} + \cdots + 5) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 7q−3q3+6q4−4q5−2q6−3q8+12q9−8q10−15q12−18q13−3q15+8q16−14q17−29q18−11q19−8q20−10q22−7q23+24q24+⋯+46q99+O(q100) 7 q - 3 q^{3} + 6 q^{4} - 4 q^{5} - 2 q^{6} - 3 q^{8} + 12 q^{9} - 8 q^{10} - 15 q^{12} - 18 q^{13} - 3 q^{15} + 8 q^{16} - 14 q^{17} - 29 q^{18} - 11 q^{19} - 8 q^{20} - 10 q^{22} - 7 q^{23} + 24 q^{24}+ \cdots + 46 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x7−10x5−x4+25x3+x2−8x+1 x^{7} - 10x^{5} - x^{4} + 25x^{3} + x^{2} - 8x + 1 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν5−2ν4−5ν3+9ν2+ν−1 \nu^{5} - 2\nu^{4} - 5\nu^{3} + 9\nu^{2} + \nu - 1 Copy content Toggle raw display
β3\beta_{3}== ν5−2ν4−6ν3+10ν2+5ν−4 \nu^{5} - 2\nu^{4} - 6\nu^{3} + 10\nu^{2} + 5\nu - 4 Copy content Toggle raw display
β4\beta_{4}== ν6−ν5−7ν4+3ν3+10ν2+4ν−2 \nu^{6} - \nu^{5} - 7\nu^{4} + 3\nu^{3} + 10\nu^{2} + 4\nu - 2 Copy content Toggle raw display
β5\beta_{5}== ν6−2ν5−6ν4+10ν3+6ν2−5ν−2 \nu^{6} - 2\nu^{5} - 6\nu^{4} + 10\nu^{3} + 6\nu^{2} - 5\nu - 2 Copy content Toggle raw display
β6\beta_{6}== −ν6+ν5+8ν4−4ν3−15ν2−ν+3 -\nu^{6} + \nu^{5} + 8\nu^{4} - 4\nu^{3} - 15\nu^{2} - \nu + 3 Copy content Toggle raw display

Display νj\nu^j in terms of βi\beta_i

ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β6+β5+β3+β1+3 \beta_{6} + \beta_{5} + \beta_{3} + \beta _1 + 3 Copy content Toggle raw display
ν3\nu^{3}== β6+β5+β2+5β1 \beta_{6} + \beta_{5} + \beta_{2} + 5\beta_1 Copy content Toggle raw display
ν4\nu^{4}== 7β6+6β5+β4+5β3+β2+7β1+14 7\beta_{6} + 6\beta_{5} + \beta_{4} + 5\beta_{3} + \beta_{2} + 7\beta _1 + 14 Copy content Toggle raw display
ν5\nu^{5}== 10β6+8β5+2β4+β3+8β2+29β1+2 10\beta_{6} + 8\beta_{5} + 2\beta_{4} + \beta_{3} + 8\beta_{2} + 29\beta _1 + 2 Copy content Toggle raw display
ν6\nu^{6}== 46β6+37β5+10β4+26β3+12β2+49β1+72 46\beta_{6} + 37\beta_{5} + 10\beta_{4} + 26\beta_{3} + 12\beta_{2} + 49\beta _1 + 72 Copy content Toggle raw display

Display βi\beta_i in terms of νj\nu^j

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
2.61328
1.76233
0.499048
0.134836
−0.692701
−2.14090
−2.17588
−2.61328 −3.04376 4.82921 1.44516 7.95419 0 −7.39352 6.26448 −3.77661
1.2 −1.76233 3.03433 1.10579 −1.94522 −5.34747 0 1.57589 6.20713 3.42812
1.3 −0.499048 1.22023 −1.75095 0.0259569 −0.608953 0 1.87191 −1.51104 −0.0129537
1.4 −0.134836 −2.58533 −1.98182 −1.71441 0.348596 0 0.536894 3.68392 0.231165
1.5 0.692701 0.603330 −1.52016 2.66776 0.417927 0 −2.43842 −2.63599 1.84796
1.6 2.14090 −2.43874 2.58346 −0.818530 −5.22109 0 1.24912 2.94743 −1.75239
1.7 2.17588 0.209939 2.73447 −3.66071 0.456804 0 1.59813 −2.95593 −7.96529
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
77 +1 +1
2323 +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 1127.2.a.l 7
7.b odd 2 1 1127.2.a.m 7
7.c even 3 2 161.2.e.b ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
161.2.e.b ✓ 14 7.c even 3 2
1127.2.a.l 7 1.a even 1 1 trivial
1127.2.a.m 7 7.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 S2new(Γ0(1127))S_{2}^{\mathrm{new}}(\Gamma_0(1127)):

T27−10T25+T24+25T23−T22−8T2−1 T_{2}^{7} - 10T_{2}^{5} + T_{2}^{4} + 25T_{2}^{3} - T_{2}^{2} - 8T_{2} - 1 Copy content Toggle raw display
T37+3T36−12T35−35T34+32T33+67T32−58T3+9 T_{3}^{7} + 3T_{3}^{6} - 12T_{3}^{5} - 35T_{3}^{4} + 32T_{3}^{3} + 67T_{3}^{2} - 58T_{3} + 9 Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T7−10T5+⋯−1 T^{7} - 10 T^{5} + \cdots - 1 Copy content Toggle raw display
33 T7+3T6+⋯+9 T^{7} + 3 T^{6} + \cdots + 9 Copy content Toggle raw display
55 T7+4T6+⋯−1 T^{7} + 4 T^{6} + \cdots - 1 Copy content Toggle raw display
77 T7 T^{7} Copy content Toggle raw display
1111 T7−25T5+⋯+1 T^{7} - 25 T^{5} + \cdots + 1 Copy content Toggle raw display
1313 T7+18T6+⋯−183 T^{7} + 18 T^{6} + \cdots - 183 Copy content Toggle raw display
1717 T7+14T6+⋯−369 T^{7} + 14 T^{6} + \cdots - 369 Copy content Toggle raw display
1919 T7+11T6+⋯−7 T^{7} + 11 T^{6} + \cdots - 7 Copy content Toggle raw display
2323 (T+1)7 (T + 1)^{7} Copy content Toggle raw display
2929 T7−3T6+⋯−73159 T^{7} - 3 T^{6} + \cdots - 73159 Copy content Toggle raw display
3131 T7+6T6+⋯−34227 T^{7} + 6 T^{6} + \cdots - 34227 Copy content Toggle raw display
3737 T7+T6+⋯−46273 T^{7} + T^{6} + \cdots - 46273 Copy content Toggle raw display
4141 T7+9T6+⋯−19917 T^{7} + 9 T^{6} + \cdots - 19917 Copy content Toggle raw display
4343 T7−9T6+⋯−78303 T^{7} - 9 T^{6} + \cdots - 78303 Copy content Toggle raw display
4747 T7−12T6+⋯−28131 T^{7} - 12 T^{6} + \cdots - 28131 Copy content Toggle raw display
5353 T7+12T6+⋯+223 T^{7} + 12 T^{6} + \cdots + 223 Copy content Toggle raw display
5959 T7−7T6+⋯−6147 T^{7} - 7 T^{6} + \cdots - 6147 Copy content Toggle raw display
6161 T7+24T6+⋯−59067 T^{7} + 24 T^{6} + \cdots - 59067 Copy content Toggle raw display
6767 T7+12T6+⋯+6033 T^{7} + 12 T^{6} + \cdots + 6033 Copy content Toggle raw display
7171 T7−20T6+⋯−488617 T^{7} - 20 T^{6} + \cdots - 488617 Copy content Toggle raw display
7373 T7+33T6+⋯+2763 T^{7} + 33 T^{6} + \cdots + 2763 Copy content Toggle raw display
7979 T7+10T6+⋯+387809 T^{7} + 10 T^{6} + \cdots + 387809 Copy content Toggle raw display
8383 T7+25T6+⋯+911 T^{7} + 25 T^{6} + \cdots + 911 Copy content Toggle raw display
8989 T7+11T6+⋯−24873 T^{7} + 11 T^{6} + \cdots - 24873 Copy content Toggle raw display
9797 T7+51T6+⋯+805725 T^{7} + 51 T^{6} + \cdots + 805725 Copy content Toggle raw display
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