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Properties

Label 171.10.a.e
Level 171171
Weight 1010
Character orbit 171.a
Self dual yes
Analytic conductor 88.07188.071
Analytic rank 00
Dimension 88
CM no
Inner twists 11

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [171,10,Mod(1,171)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(171, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 10, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("171.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Level: N N == 171=32⋅19 171 = 3^{2} \cdot 19
Weight: k k == 10 10
Character orbit: [χ][\chi] == 171.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-17] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 88.071127984088.0711279840
Analytic rank: 00
Dimension: 88
Coefficient field: Q[x]/(x8−⋯ )\mathbb{Q}[x]/(x^{8} - \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x8−x7−3446x6+2146x5+3632756x4+1877896x3−1128074928x2+⋯−684004608 x^{8} - x^{7} - 3446 x^{6} + 2146 x^{5} + 3632756 x^{4} + 1877896 x^{3} - 1128074928 x^{2} + \cdots - 684004608 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 25⋅32 2^{5}\cdot 3^{2}
Twist minimal: no (minimal twist has level 57)
Fricke sign: −1-1
Sato-Tate group: SU(2)\mathrm{SU}(2)

qq-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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+4β1+354)q4+(−β4+β2−15β1−485)q5+(β7+2β4+β3+⋯+1181)q7+(−β7−3β6+⋯−3451)q8+⋯+(48416β7+192055β6+⋯+634215594)q98+O(q100) q + ( - \beta_1 - 2) q^{2} + (\beta_{2} + 4 \beta_1 + 354) q^{4} + ( - \beta_{4} + \beta_{2} - 15 \beta_1 - 485) q^{5} + (\beta_{7} + 2 \beta_{4} + \beta_{3} + \cdots + 1181) q^{7} + ( - \beta_{7} - 3 \beta_{6} + \cdots - 3451) q^{8}+ \cdots + (48416 \beta_{7} + 192055 \beta_{6} + \cdots + 634215594) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 8q−17q2+2833q4−3902q5+9488q7−27927q8+111324q10−38328q11+238594q13−255570q14+875017q16−340248q17−1042568q19+70298q20−2034178q22+⋯+5080621865q98+O(q100) 8 q - 17 q^{2} + 2833 q^{4} - 3902 q^{5} + 9488 q^{7} - 27927 q^{8} + 111324 q^{10} - 38328 q^{11} + 238594 q^{13} - 255570 q^{14} + 875017 q^{16} - 340248 q^{17} - 1042568 q^{19} + 70298 q^{20} - 2034178 q^{22}+ \cdots + 5080621865 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8−x7−3446x6+2146x5+3632756x4+1877896x3−1128074928x2+⋯−684004608 x^{8} - x^{7} - 3446 x^{6} + 2146 x^{5} + 3632756 x^{4} + 1877896 x^{3} - 1128074928 x^{2} + \cdots - 684004608 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν2−862 \nu^{2} - 862 Copy content Toggle raw display
β3\beta_{3}== (−248545ν7−6574249ν6+755626284ν5+16477812710ν4+⋯+919755537257088)/328794036480 ( - 248545 \nu^{7} - 6574249 \nu^{6} + 755626284 \nu^{5} + 16477812710 \nu^{4} + \cdots + 919755537257088 ) / 328794036480 Copy content Toggle raw display
β4\beta_{4}== (680653ν7−3581003ν6−2400722940ν5+8005981330ν4+⋯−11 ⁣⋯ ⁣40)/328794036480 ( 680653 \nu^{7} - 3581003 \nu^{6} - 2400722940 \nu^{5} + 8005981330 \nu^{4} + \cdots - 11\!\cdots\!40 ) / 328794036480 Copy content Toggle raw display
β5\beta_{5}== (−64873ν7+144686ν6+236289207ν5−526583290ν4+⋯+45613645265664)/13699751520 ( - 64873 \nu^{7} + 144686 \nu^{6} + 236289207 \nu^{5} - 526583290 \nu^{4} + \cdots + 45613645265664 ) / 13699751520 Copy content Toggle raw display
β6\beta_{6}== (−1841779ν7−1545259ν6+6239657508ν5+1122039890ν4+⋯+19 ⁣⋯ ⁣96)/328794036480 ( - 1841779 \nu^{7} - 1545259 \nu^{6} + 6239657508 \nu^{5} + 1122039890 \nu^{4} + \cdots + 19\!\cdots\!96 ) / 328794036480 Copy content Toggle raw display
β7\beta_{7}== (35797ν7−85625ν6−114994362ν5+334949170ν4+111824975932ν3+⋯−15190761446784)/6322962240 ( 35797 \nu^{7} - 85625 \nu^{6} - 114994362 \nu^{5} + 334949170 \nu^{4} + 111824975932 \nu^{3} + \cdots - 15190761446784 ) / 6322962240 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}== β2+862 \beta_{2} + 862 Copy content Toggle raw display
ν3\nu^{3}== β7+3β6−β5+2β4−3β3−4β2+1338β1+319 \beta_{7} + 3\beta_{6} - \beta_{5} + 2\beta_{4} - 3\beta_{3} - 4\beta_{2} + 1338\beta _1 + 319 Copy content Toggle raw display
ν4\nu^{4}== 7β7−27β6−23β5−124β4+57β3+1742β2−1606β1+1153903 7\beta_{7} - 27\beta_{6} - 23\beta_{5} - 124\beta_{4} + 57\beta_{3} + 1742\beta_{2} - 1606\beta _1 + 1153903 Copy content Toggle raw display
ν5\nu^{5}== 2729β7+7055β6−1801β5+4904β4−7129β3−11300β2+⋯−892147 2729 \beta_{7} + 7055 \beta_{6} - 1801 \beta_{5} + 4904 \beta_{4} - 7129 \beta_{3} - 11300 \beta_{2} + \cdots - 892147 Copy content Toggle raw display
ν6\nu^{6}== 2649β7−92825β6−56281β5−341116β4+126091β3+⋯+1689294713 2649 \beta_{7} - 92825 \beta_{6} - 56281 \beta_{5} - 341116 \beta_{4} + 126091 \beta_{3} + \cdots + 1689294713 Copy content Toggle raw display
ν7\nu^{7}== 5760253β7+13322507β6−2581085β5+9850212β4−13761361β3+⋯−4603466083 5760253 \beta_{7} + 13322507 \beta_{6} - 2581085 \beta_{5} + 9850212 \beta_{4} - 13761361 \beta_{3} + \cdots - 4603466083 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
39.4843
36.8155
24.5052
−0.355238
−1.72136
−22.5401
−33.1038
−42.0845
−41.4843 0 1208.94 741.508 0 4318.56 −28912.2 0 −30760.9
1.2 −38.8155 0 994.645 −2396.22 0 3937.20 −18734.1 0 93010.5
1.3 −26.5052 0 190.524 −1296.92 0 −3384.63 8520.78 0 34375.1
1.4 −1.64476 0 −509.295 1313.38 0 −2188.42 1679.79 0 −2160.20
1.5 −0.278642 0 −511.922 −1999.74 0 10192.2 285.307 0 557.212
1.6 20.5401 0 −90.1033 −849.287 0 −10981.2 −12367.3 0 −17444.5
1.7 31.1038 0 455.444 −1145.37 0 10346.9 −1759.12 0 −35625.3
1.8 40.0845 0 1094.76 1730.65 0 −2752.52 23359.8 0 69372.1
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
33 −1 -1
1919 +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 171.10.a.e 8
3.b odd 2 1 57.10.a.d ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.10.a.d ✓ 8 3.b odd 2 1
171.10.a.e 8 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T28+17T27−3320T26−42966T25+3405936T24+26549304T23+⋯−500910592 T_{2}^{8} + 17 T_{2}^{7} - 3320 T_{2}^{6} - 42966 T_{2}^{5} + 3405936 T_{2}^{4} + 26549304 T_{2}^{3} + \cdots - 500910592 acting on S10new(Γ0(171))S_{10}^{\mathrm{new}}(\Gamma_0(171)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T8+17T7+⋯−500910592 T^{8} + 17 T^{7} + \cdots - 500910592 Copy content Toggle raw display
33 T8 T^{8} Copy content Toggle raw display
55 T8+⋯−10 ⁣⋯ ⁣00 T^{8} + \cdots - 10\!\cdots\!00 Copy content Toggle raw display
77 T8+⋯+40 ⁣⋯ ⁣00 T^{8} + \cdots + 40\!\cdots\!00 Copy content Toggle raw display
1111 T8+⋯+25 ⁣⋯ ⁣00 T^{8} + \cdots + 25\!\cdots\!00 Copy content Toggle raw display
1313 T8+⋯−95 ⁣⋯ ⁣08 T^{8} + \cdots - 95\!\cdots\!08 Copy content Toggle raw display
1717 T8+⋯+48 ⁣⋯ ⁣12 T^{8} + \cdots + 48\!\cdots\!12 Copy content Toggle raw display
1919 (T+130321)8 (T + 130321)^{8} Copy content Toggle raw display
2323 T8+⋯−74 ⁣⋯ ⁣04 T^{8} + \cdots - 74\!\cdots\!04 Copy content Toggle raw display
2929 T8+⋯−12 ⁣⋯ ⁣00 T^{8} + \cdots - 12\!\cdots\!00 Copy content Toggle raw display
3131 T8+⋯−11 ⁣⋯ ⁣28 T^{8} + \cdots - 11\!\cdots\!28 Copy content Toggle raw display
3737 T8+⋯+22 ⁣⋯ ⁣00 T^{8} + \cdots + 22\!\cdots\!00 Copy content Toggle raw display
4141 T8+⋯−12 ⁣⋯ ⁣36 T^{8} + \cdots - 12\!\cdots\!36 Copy content Toggle raw display
4343 T8+⋯+53 ⁣⋯ ⁣36 T^{8} + \cdots + 53\!\cdots\!36 Copy content Toggle raw display
4747 T8+⋯+26 ⁣⋯ ⁣72 T^{8} + \cdots + 26\!\cdots\!72 Copy content Toggle raw display
5353 T8+⋯+94 ⁣⋯ ⁣72 T^{8} + \cdots + 94\!\cdots\!72 Copy content Toggle raw display
5959 T8+⋯−13 ⁣⋯ ⁣00 T^{8} + \cdots - 13\!\cdots\!00 Copy content Toggle raw display
6161 T8+⋯+64 ⁣⋯ ⁣88 T^{8} + \cdots + 64\!\cdots\!88 Copy content Toggle raw display
6767 T8+⋯−18 ⁣⋯ ⁣56 T^{8} + \cdots - 18\!\cdots\!56 Copy content Toggle raw display
7171 T8+⋯−72 ⁣⋯ ⁣00 T^{8} + \cdots - 72\!\cdots\!00 Copy content Toggle raw display
7373 T8+⋯−67 ⁣⋯ ⁣00 T^{8} + \cdots - 67\!\cdots\!00 Copy content Toggle raw display
7979 T8+⋯−76 ⁣⋯ ⁣00 T^{8} + \cdots - 76\!\cdots\!00 Copy content Toggle raw display
8383 T8+⋯−36 ⁣⋯ ⁣16 T^{8} + \cdots - 36\!\cdots\!16 Copy content Toggle raw display
8989 T8+⋯−28 ⁣⋯ ⁣00 T^{8} + \cdots - 28\!\cdots\!00 Copy content Toggle raw display
9797 T8+⋯+10 ⁣⋯ ⁣00 T^{8} + \cdots + 10\!\cdots\!00 Copy content Toggle raw display
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