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Properties

Label 175.8.a.i
Level 175175
Weight 88
Character orbit 175.a
Self dual yes
Analytic conductor 54.66754.667
Analytic rank 00
Dimension 88
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] = [175,8,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, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("175.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: N N == 175=52⋅7 175 = 5^{2} \cdot 7
Weight: k k == 8 8
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: 54.667379459754.6673794597
Analytic rank: 00
Dimension: 88
Coefficient field: Q[x]/(x8−⋯ )\mathbb{Q}[x]/(x^{8} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x8−2x7−706x6+1459x5+140610x4−269712x3−8960496x2+18558112x+98712192 x^{8} - 2x^{7} - 706x^{6} + 1459x^{5} + 140610x^{4} - 269712x^{3} - 8960496x^{2} + 18558112x + 98712192 Copy content Toggle raw display
Coefficient ring: Z[a1,a2,a3]\Z[a_1, a_2, a_3]
Coefficient ring index: 24⋅53 2^{4}\cdot 5^{3}
Twist minimal: yes
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,…,β71,\beta_1,\ldots,\beta_{7} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(−β1−2)q2+(−β2−7)q3+(β3+β2+4β1+53)q4+(β5−β3+2β2+⋯−14)q6−343q7+(−β7−2β5+⋯−463)q8+⋯+(−2889β7+3525β6+⋯+1420540)q99+O(q100) q + ( - \beta_1 - 2) q^{2} + ( - \beta_{2} - 7) q^{3} + (\beta_{3} + \beta_{2} + 4 \beta_1 + 53) q^{4} + (\beta_{5} - \beta_{3} + 2 \beta_{2} + \cdots - 14) q^{6} - 343 q^{7} + ( - \beta_{7} - 2 \beta_{5} + \cdots - 463) q^{8}+ \cdots + ( - 2889 \beta_{7} + 3525 \beta_{6} + \cdots + 1420540) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 8q−18q2−54q3+432q4−53q6−2744q7−3843q8+12334q9−672q11−37071q12+9410q13+6174q14+39152q16+34626q17−1861q18−1876q19+⋯+11403964q99+O(q100) 8 q - 18 q^{2} - 54 q^{3} + 432 q^{4} - 53 q^{6} - 2744 q^{7} - 3843 q^{8} + 12334 q^{9} - 672 q^{11} - 37071 q^{12} + 9410 q^{13} + 6174 q^{14} + 39152 q^{16} + 34626 q^{17} - 1861 q^{18} - 1876 q^{19}+ \cdots + 11403964 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8−2x7−706x6+1459x5+140610x4−269712x3−8960496x2+18558112x+98712192 x^{8} - 2x^{7} - 706x^{6} + 1459x^{5} + 140610x^{4} - 269712x^{3} - 8960496x^{2} + 18558112x + 98712192 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (29ν7−9230ν6+41966ν5+5560783ν4−30942434ν3+⋯+11004010176)/77323680 ( 29 \nu^{7} - 9230 \nu^{6} + 41966 \nu^{5} + 5560783 \nu^{4} - 30942434 \nu^{3} + \cdots + 11004010176 ) / 77323680 Copy content Toggle raw display
β3\beta_{3}== (−29ν7+9230ν6−41966ν5−5560783ν4+30942434ν3+⋯−24690301536)/77323680 ( - 29 \nu^{7} + 9230 \nu^{6} - 41966 \nu^{5} - 5560783 \nu^{4} + 30942434 \nu^{3} + \cdots - 24690301536 ) / 77323680 Copy content Toggle raw display
β4\beta_{4}== (−43ν7+8131ν6−40006ν5−4456883ν4+33265069ν3+⋯+733313856)/1933092 ( - 43 \nu^{7} + 8131 \nu^{6} - 40006 \nu^{5} - 4456883 \nu^{4} + 33265069 \nu^{3} + \cdots + 733313856 ) / 1933092 Copy content Toggle raw display
β5\beta_{5}== (−3067ν7+23890ν6+1825502ν5−13526969ν4−224298338ν3+⋯−8462630688)/25774560 ( - 3067 \nu^{7} + 23890 \nu^{6} + 1825502 \nu^{5} - 13526969 \nu^{4} - 224298338 \nu^{3} + \cdots - 8462630688 ) / 25774560 Copy content Toggle raw display
β6\beta_{6}== (−247ν7−212ν6+141974ν5+137359ν4−16174868ν3+⋯+1781878656)/1288728 ( - 247 \nu^{7} - 212 \nu^{6} + 141974 \nu^{5} + 137359 \nu^{4} - 16174868 \nu^{3} + \cdots + 1781878656 ) / 1288728 Copy content Toggle raw display
β7\beta_{7}== (6163ν7−57010ν6−3609038ν5+32614721ν4+443428802ν3+⋯+30377854752)/25774560 ( 6163 \nu^{7} - 57010 \nu^{6} - 3609038 \nu^{5} + 32614721 \nu^{4} + 443428802 \nu^{3} + \cdots + 30377854752 ) / 25774560 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}== β3+β2+177 \beta_{3} + \beta_{2} + 177 Copy content Toggle raw display
ν3\nu^{3}== β7+2β5−3β2+306β1−95 \beta_{7} + 2\beta_{5} - 3\beta_{2} + 306\beta _1 - 95 Copy content Toggle raw display
ν4\nu^{4}== −β7−2β6+β5+4β4+428β3+598β2−67β1+54318 -\beta_{7} - 2\beta_{6} + \beta_{5} + 4\beta_{4} + 428\beta_{3} + 598\beta_{2} - 67\beta _1 + 54318 Copy content Toggle raw display
ν5\nu^{5}== 513β7−61β6+1128β5−13β4−406β3−1527β2+113619β1−57319 513\beta_{7} - 61\beta_{6} + 1128\beta_{5} - 13\beta_{4} - 406\beta_{3} - 1527\beta_{2} + 113619\beta _1 - 57319 Copy content Toggle raw display
ν6\nu^{6}== −901β7−1608β6+652β5+2328β4+177952β3+275581β2+⋯+20220017 - 901 \beta_{7} - 1608 \beta_{6} + 652 \beta_{5} + 2328 \beta_{4} + 177952 \beta_{3} + 275581 \beta_{2} + \cdots + 20220017 Copy content Toggle raw display
ν7\nu^{7}== 229601β7−40012β6+517393β5−7246β4−304168β3+⋯−34285736 229601 \beta_{7} - 40012 \beta_{6} + 517393 \beta_{5} - 7246 \beta_{4} - 304168 \beta_{3} + \cdots - 34285736 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
20.2756
13.9780
8.63295
5.57628
−2.61316
−11.2353
−11.8923
−20.7221
−22.2756 −63.0794 368.203 0 1405.13 −343.000 −5350.66 1792.01 0
1.2 −15.9780 17.4271 127.297 0 −278.451 −343.000 11.2405 −1883.29 0
1.3 −10.6329 82.7707 −14.9404 0 −880.097 −343.000 1519.88 4663.99 0
1.4 −7.57628 −78.3569 −70.6000 0 593.654 −343.000 1504.65 3952.80 0
1.5 0.613157 4.22092 −127.624 0 2.58809 −343.000 −156.738 −2169.18 0
1.6 9.23526 79.3182 −42.7100 0 732.524 −343.000 −1576.55 4104.37 0
1.7 9.89229 −19.7743 −30.1426 0 −195.613 −343.000 −1564.39 −1795.98 0
1.8 18.7221 −76.5263 222.518 0 −1432.74 −343.000 1769.57 3669.28 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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 175.8.a.i ✓ 8
5.b even 2 1 175.8.a.j yes 8
5.c odd 4 2 175.8.b.h 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
175.8.a.i ✓ 8 1.a even 1 1 trivial
175.8.a.j yes 8 5.b even 2 1
175.8.b.h 16 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T28+18T27−566T26−9315T25+85340T24+1226184T23+⋯+30070080 T_{2}^{8} + 18 T_{2}^{7} - 566 T_{2}^{6} - 9315 T_{2}^{5} + 85340 T_{2}^{4} + 1226184 T_{2}^{3} + \cdots + 30070080 acting on S8new(Γ0(175))S_{8}^{\mathrm{new}}(\Gamma_0(175)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T8+18T7+⋯+30070080 T^{8} + 18 T^{7} + \cdots + 30070080 Copy content Toggle raw display
33 T8+⋯+3612097105344 T^{8} + \cdots + 3612097105344 Copy content Toggle raw display
55 T8 T^{8} Copy content Toggle raw display
77 (T+343)8 (T + 343)^{8} Copy content Toggle raw display
1111 T8+⋯−42 ⁣⋯ ⁣31 T^{8} + \cdots - 42\!\cdots\!31 Copy content Toggle raw display
1313 T8+⋯+54 ⁣⋯ ⁣00 T^{8} + \cdots + 54\!\cdots\!00 Copy content Toggle raw display
1717 T8+⋯+38 ⁣⋯ ⁣68 T^{8} + \cdots + 38\!\cdots\!68 Copy content Toggle raw display
1919 T8+⋯+14 ⁣⋯ ⁣00 T^{8} + \cdots + 14\!\cdots\!00 Copy content Toggle raw display
2323 T8+⋯−14 ⁣⋯ ⁣56 T^{8} + \cdots - 14\!\cdots\!56 Copy content Toggle raw display
2929 T8+⋯−31 ⁣⋯ ⁣00 T^{8} + \cdots - 31\!\cdots\!00 Copy content Toggle raw display
3131 T8+⋯−16 ⁣⋯ ⁣00 T^{8} + \cdots - 16\!\cdots\!00 Copy content Toggle raw display
3737 T8+⋯−44 ⁣⋯ ⁣84 T^{8} + \cdots - 44\!\cdots\!84 Copy content Toggle raw display
4141 T8+⋯+33 ⁣⋯ ⁣44 T^{8} + \cdots + 33\!\cdots\!44 Copy content Toggle raw display
4343 T8+⋯−35 ⁣⋯ ⁣00 T^{8} + \cdots - 35\!\cdots\!00 Copy content Toggle raw display
4747 T8+⋯−26 ⁣⋯ ⁣44 T^{8} + \cdots - 26\!\cdots\!44 Copy content Toggle raw display
5353 T8+⋯−17 ⁣⋯ ⁣08 T^{8} + \cdots - 17\!\cdots\!08 Copy content Toggle raw display
5959 T8+⋯−31 ⁣⋯ ⁣00 T^{8} + \cdots - 31\!\cdots\!00 Copy content Toggle raw display
6161 T8+⋯+40 ⁣⋯ ⁣36 T^{8} + \cdots + 40\!\cdots\!36 Copy content Toggle raw display
6767 T8+⋯+89 ⁣⋯ ⁣13 T^{8} + \cdots + 89\!\cdots\!13 Copy content Toggle raw display
7171 T8+⋯−86 ⁣⋯ ⁣20 T^{8} + \cdots - 86\!\cdots\!20 Copy content Toggle raw display
7373 T8+⋯+17 ⁣⋯ ⁣64 T^{8} + \cdots + 17\!\cdots\!64 Copy content Toggle raw display
7979 T8+⋯−26 ⁣⋯ ⁣00 T^{8} + \cdots - 26\!\cdots\!00 Copy content Toggle raw display
8383 T8+⋯−10 ⁣⋯ ⁣52 T^{8} + \cdots - 10\!\cdots\!52 Copy content Toggle raw display
8989 T8+⋯−31 ⁣⋯ ⁣00 T^{8} + \cdots - 31\!\cdots\!00 Copy content Toggle raw display
9797 T8+⋯−56 ⁣⋯ ⁣32 T^{8} + \cdots - 56\!\cdots\!32 Copy content Toggle raw display
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