Properties

Label 57.10.a.d
Level 5757
Weight 1010
Character orbit 57.a
Self dual yes
Analytic conductor 29.35729.357
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] = [57,10,Mod(1,57)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(57, 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("57.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: N N == 57=319 57 = 3 \cdot 19
Weight: k k == 10 10
Character orbit: [χ][\chi] == 57.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: 29.357042661329.3570426613
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: x8x73446x6+2146x5+3632756x4+1877896x31128074928x2+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: 2532 2^{5}\cdot 3^{2}
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+81q3+(β2+4β1+354)q4+(β4β2+15β1+485)q5+(81β1+162)q6+(β7+2β4+β3++1181)q7++(19683β745927β6++31676508)q99+O(q100) q + (\beta_1 + 2) q^{2} + 81 q^{3} + (\beta_{2} + 4 \beta_1 + 354) q^{4} + (\beta_{4} - \beta_{2} + 15 \beta_1 + 485) q^{5} + (81 \beta_1 + 162) q^{6} + (\beta_{7} + 2 \beta_{4} + \beta_{3} + \cdots + 1181) q^{7}+ \cdots + (19683 \beta_{7} - 45927 \beta_{6} + \cdots + 31676508) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 8q+17q2+648q3+2833q4+3902q5+1377q6+9488q7+27927q8+52488q9+111324q10+38328q11+229473q12+238594q13+255570q14+316062q15++251470008q99+O(q100) 8 q + 17 q^{2} + 648 q^{3} + 2833 q^{4} + 3902 q^{5} + 1377 q^{6} + 9488 q^{7} + 27927 q^{8} + 52488 q^{9} + 111324 q^{10} + 38328 q^{11} + 229473 q^{12} + 238594 q^{13} + 255570 q^{14} + 316062 q^{15}+ \cdots + 251470008 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x8x73446x6+2146x5+3632756x4+1877896x31128074928x2+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}== ν2862 \nu^{2} - 862 Copy content Toggle raw display
β3\beta_{3}== (248545ν76574249ν6+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}== (680653ν73581003ν62400722940ν5+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+236289207ν5526583290ν4++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}== (1841779ν71545259ν6+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}== (35797ν785625ν6114994362ν5+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
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+3β6β5+2β43β34β2+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}== 7β727β623β5124β4+57β3+1742β21606β1+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+7055β61801β5+4904β47129β311300β2+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}== 2649β792825β656281β5341116β4+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+13322507β62581085β5+9850212β413761361β3+4603466083 5760253 \beta_{7} + 13322507 \beta_{6} - 2581085 \beta_{5} + 9850212 \beta_{4} - 13761361 \beta_{3} + \cdots - 4603466083 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
−42.0845
−33.1038
−22.5401
−1.72136
−0.355238
24.5052
36.8155
39.4843
−40.0845 81.0000 1094.76 −1730.65 −3246.84 −2752.52 −23359.8 6561.00 69372.1
1.2 −31.1038 81.0000 455.444 1145.37 −2519.40 10346.9 1759.12 6561.00 −35625.3
1.3 −20.5401 81.0000 −90.1033 849.287 −1663.75 −10981.2 12367.3 6561.00 −17444.5
1.4 0.278642 81.0000 −511.922 1999.74 22.5700 10192.2 −285.307 6561.00 557.212
1.5 1.64476 81.0000 −509.295 −1313.38 133.226 −2188.42 −1679.79 6561.00 −2160.20
1.6 26.5052 81.0000 190.524 1296.92 2146.92 −3384.63 −8520.78 6561.00 34375.1
1.7 38.8155 81.0000 994.645 2396.22 3144.06 3937.20 18734.1 6561.00 93010.5
1.8 41.4843 81.0000 1208.94 −741.508 3360.23 4318.56 28912.2 6561.00 −30760.9
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 57.10.a.d 8
3.b odd 2 1 171.10.a.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.10.a.d 8 1.a even 1 1 trivial
171.10.a.e 8 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2817T273320T26+42966T25+3405936T2426549304T23+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(57))S_{10}^{\mathrm{new}}(\Gamma_0(57)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T817T7+500910592 T^{8} - 17 T^{7} + \cdots - 500910592 Copy content Toggle raw display
33 (T81)8 (T - 81)^{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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