Properties

Label 175.10.a.e
Level 175175
Weight 1010
Character orbit 175.a
Self dual yes
Analytic conductor 90.13190.131
Analytic rank 00
Dimension 44
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] = [175,10,Mod(1,175)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(175, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: N N == 175=527 175 = 5^{2} \cdot 7
Weight: k k == 10 10
Character orbit: [χ][\chi] == 175.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: 90.131271328790.1312713287
Analytic rank: 00
Dimension: 44
Coefficient field: Q[x]/(x4)\mathbb{Q}[x]/(x^{4} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x4x3648x2+6926x8308 x^{4} - x^{3} - 648x^{2} + 6926x - 8308 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 2325 2\cdot 3^{2}\cdot 5
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 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+(β1+5)q2+(β2+4)q3+(β3+β2+9β1+435)q4+(7β319β2+48)q6+2401q7+(7β3+95β2++7725)q8++(249624β3++303476508)q99+O(q100) q + (\beta_1 + 5) q^{2} + ( - \beta_{2} + 4) q^{3} + (\beta_{3} + \beta_{2} + 9 \beta_1 + 435) q^{4} + ( - 7 \beta_{3} - 19 \beta_{2} + \cdots - 48) q^{6} + 2401 q^{7} + (7 \beta_{3} + 95 \beta_{2} + \cdots + 7725) q^{8}+ \cdots + ( - 249624 \beta_{3} + \cdots + 303476508) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q+19q2+18q3+1729q4144q6+9604q7+30495q8+5382q9+82438q1141328q12+72962q13+45619q14+64257q16+357542q17+965367q18+300732q19++1222369524q99+O(q100) 4 q + 19 q^{2} + 18 q^{3} + 1729 q^{4} - 144 q^{6} + 9604 q^{7} + 30495 q^{8} + 5382 q^{9} + 82438 q^{11} - 41328 q^{12} + 72962 q^{13} + 45619 q^{14} + 64257 q^{16} + 357542 q^{17} + 965367 q^{18} + 300732 q^{19}+ \cdots + 1222369524 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x4x3648x2+6926x8308 x^{4} - x^{3} - 648x^{2} + 6926x - 8308 : Copy content Toggle raw display

β1\beta_{1}== (3ν347ν2+1318ν+756)/118 ( -3\nu^{3} - 47\nu^{2} + 1318\nu + 756 ) / 118 Copy content Toggle raw display
β2\beta_{2}== (15ν3117ν2+7770ν34688)/118 ( -15\nu^{3} - 117\nu^{2} + 7770\nu - 34688 ) / 118 Copy content Toggle raw display
β3\beta_{3}== (3ν3+71ν2+6038ν38656)/118 ( -3\nu^{3} + 71\nu^{2} + 6038\nu - 38656 ) / 118 Copy content Toggle raw display
ν\nu== (β3β2+4β1+8)/30 ( \beta_{3} - \beta_{2} + 4\beta _1 + 8 ) / 30 Copy content Toggle raw display
ν2\nu^{2}== (β3+4β219β1+970)/3 ( -\beta_{3} + 4\beta_{2} - 19\beta _1 + 970 ) / 3 Copy content Toggle raw display
ν3\nu^{3}== (298β3533β2+1777β170446)/15 ( 298\beta_{3} - 533\beta_{2} + 1777\beta _1 - 70446 ) / 15 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
16.7769
−29.3917
1.37673
12.2380
−33.3659 72.5961 601.285 0 −2422.24 2401.00 −2979.08 −14412.8 0
1.2 −15.4436 −137.736 −273.496 0 2127.13 2401.00 12130.9 −711.820 0
1.3 25.9629 209.523 162.070 0 5439.82 2401.00 −9085.18 24216.9 0
1.4 41.8466 −126.383 1239.14 0 −5288.71 2401.00 30428.4 −3710.28 0
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 175.10.a.e 4
5.b even 2 1 35.10.a.c 4
5.c odd 4 2 175.10.b.e 8
15.d odd 2 1 315.10.a.g 4
35.c odd 2 1 245.10.a.e 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.10.a.c 4 5.b even 2 1
175.10.a.e 4 1.a even 1 1 trivial
175.10.b.e 8 5.c odd 4 2
245.10.a.e 4 35.c odd 2 1
315.10.a.g 4 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2419T231708T22+18088T2+559840 T_{2}^{4} - 19T_{2}^{3} - 1708T_{2}^{2} + 18088T_{2} + 559840 acting on S10new(Γ0(175))S_{10}^{\mathrm{new}}(\Gamma_0(175)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T419T3++559840 T^{4} - 19 T^{3} + \cdots + 559840 Copy content Toggle raw display
33 T418T3++264777984 T^{4} - 18 T^{3} + \cdots + 264777984 Copy content Toggle raw display
55 T4 T^{4} Copy content Toggle raw display
77 (T2401)4 (T - 2401)^{4} Copy content Toggle raw display
1111 T4++86 ⁣ ⁣76 T^{4} + \cdots + 86\!\cdots\!76 Copy content Toggle raw display
1313 T4++43 ⁣ ⁣80 T^{4} + \cdots + 43\!\cdots\!80 Copy content Toggle raw display
1717 T4+20 ⁣ ⁣04 T^{4} + \cdots - 20\!\cdots\!04 Copy content Toggle raw display
1919 T4++10 ⁣ ⁣00 T^{4} + \cdots + 10\!\cdots\!00 Copy content Toggle raw display
2323 T4++30 ⁣ ⁣16 T^{4} + \cdots + 30\!\cdots\!16 Copy content Toggle raw display
2929 T4++32 ⁣ ⁣00 T^{4} + \cdots + 32\!\cdots\!00 Copy content Toggle raw display
3131 T4+10 ⁣ ⁣72 T^{4} + \cdots - 10\!\cdots\!72 Copy content Toggle raw display
3737 T4++32 ⁣ ⁣24 T^{4} + \cdots + 32\!\cdots\!24 Copy content Toggle raw display
4141 T4+16 ⁣ ⁣76 T^{4} + \cdots - 16\!\cdots\!76 Copy content Toggle raw display
4343 T4+36 ⁣ ⁣60 T^{4} + \cdots - 36\!\cdots\!60 Copy content Toggle raw display
4747 T4+21 ⁣ ⁣12 T^{4} + \cdots - 21\!\cdots\!12 Copy content Toggle raw display
5353 T4+36 ⁣ ⁣12 T^{4} + \cdots - 36\!\cdots\!12 Copy content Toggle raw display
5959 T4++19 ⁣ ⁣00 T^{4} + \cdots + 19\!\cdots\!00 Copy content Toggle raw display
6161 T4+15 ⁣ ⁣56 T^{4} + \cdots - 15\!\cdots\!56 Copy content Toggle raw display
6767 T4++66 ⁣ ⁣12 T^{4} + \cdots + 66\!\cdots\!12 Copy content Toggle raw display
7171 T4+10 ⁣ ⁣40 T^{4} + \cdots - 10\!\cdots\!40 Copy content Toggle raw display
7373 T4++27 ⁣ ⁣68 T^{4} + \cdots + 27\!\cdots\!68 Copy content Toggle raw display
7979 T4+20 ⁣ ⁣00 T^{4} + \cdots - 20\!\cdots\!00 Copy content Toggle raw display
8383 T4+31 ⁣ ⁣52 T^{4} + \cdots - 31\!\cdots\!52 Copy content Toggle raw display
8989 T4+36 ⁣ ⁣00 T^{4} + \cdots - 36\!\cdots\!00 Copy content Toggle raw display
9797 T4++23 ⁣ ⁣68 T^{4} + \cdots + 23\!\cdots\!68 Copy content Toggle raw display
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