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Properties

Label 57.10.a.b
Level 5757
Weight 1010
Character orbit 57.a
Self dual yes
Analytic conductor 29.35729.357
Analytic rank 00
Dimension 66
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=3⋅19 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: 66
Coefficient field: Q[x]/(x6−⋯ )\mathbb{Q}[x]/(x^{6} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x6−3x5−1860x4+264x3+626016x2+4023504x−725760 x^{6} - 3x^{5} - 1860x^{4} + 264x^{3} + 626016x^{2} + 4023504x - 725760 Copy content Toggle raw display
Coefficient ring: Z[a1,…,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 23⋅33 2^{3}\cdot 3^{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,…,β51,\beta_1,\ldots,\beta_{5} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(−β1+6)q2−81q3+(β3−8β1+144)q4+(−β5+β4+29β1+179)q5+(81β1−486)q6+(−3β4+7β3+⋯+1285)q7+⋯+(91854β5+72171β4+⋯−61561863)q99+O(q100) q + ( - \beta_1 + 6) q^{2} - 81 q^{3} + (\beta_{3} - 8 \beta_1 + 144) q^{4} + ( - \beta_{5} + \beta_{4} + 29 \beta_1 + 179) q^{5} + (81 \beta_1 - 486) q^{6} + ( - 3 \beta_{4} + 7 \beta_{3} + \cdots + 1285) q^{7}+ \cdots + (91854 \beta_{5} + 72171 \beta_{4} + \cdots - 61561863) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr⁡(f)(q)\operatorname{Tr}(f)(q) == 6q+33q2−486q3+837q4+1158q5−2673q6+7890q7+18327q8+39366q9−100152q10−55530q11−67797q12+90432q13−188448q14−93798q15+474897q16+⋯−364332330q99+O(q100) 6 q + 33 q^{2} - 486 q^{3} + 837 q^{4} + 1158 q^{5} - 2673 q^{6} + 7890 q^{7} + 18327 q^{8} + 39366 q^{9} - 100152 q^{10} - 55530 q^{11} - 67797 q^{12} + 90432 q^{13} - 188448 q^{14} - 93798 q^{15} + 474897 q^{16}+ \cdots - 364332330 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x6−3x5−1860x4+264x3+626016x2+4023504x−725760 x^{6} - 3x^{5} - 1860x^{4} + 264x^{3} + 626016x^{2} + 4023504x - 725760 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (15ν5−47ν4−25530ν3+42820ν2+6429864ν+22805376)/70912 ( 15\nu^{5} - 47\nu^{4} - 25530\nu^{3} + 42820\nu^{2} + 6429864\nu + 22805376 ) / 70912 Copy content Toggle raw display
β3\beta_{3}== ν2−4ν−620 \nu^{2} - 4\nu - 620 Copy content Toggle raw display
β4\beta_{4}== (87ν5−1159ν4−152506ν3+1524772ν2+41842216ν+10203264)/70912 ( 87\nu^{5} - 1159\nu^{4} - 152506\nu^{3} + 1524772\nu^{2} + 41842216\nu + 10203264 ) / 70912 Copy content Toggle raw display
β5\beta_{5}== (ν5−77ν4−594ν3+119564ν2−988696ν−18109856)/8864 ( \nu^{5} - 77\nu^{4} - 594\nu^{3} + 119564\nu^{2} - 988696\nu - 18109856 ) / 8864 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+4β1+620 \beta_{3} + 4\beta _1 + 620 Copy content Toggle raw display
ν3\nu^{3}== 6β5−4β4−7β3+20β2+1188β1+2062 6\beta_{5} - 4\beta_{4} - 7\beta_{3} + 20\beta_{2} + 1188\beta _1 + 2062 Copy content Toggle raw display
ν4\nu^{4}== −30β5−60β4+1475β3+364β2+4952β1+744778 -30\beta_{5} - 60\beta_{4} + 1475\beta_{3} + 364\beta_{2} + 4952\beta _1 + 744778 Copy content Toggle raw display
ν5\nu^{5}== 10118β5−6996β4−10147β3+39908β2+1597416β1+2552910 10118\beta_{5} - 6996\beta_{4} - 10147\beta_{3} + 39908\beta_{2} + 1597416\beta _1 + 2552910 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
38.3673
24.3919
0.175583
−8.10134
−14.6700
−37.1634
−32.3673 −81.0000 535.640 542.116 2621.75 4515.51 −765.163 6561.00 −17546.8
1.2 −18.3919 −81.0000 −173.738 1624.78 1489.74 7425.81 12612.0 6561.00 −29882.9
1.3 5.82442 −81.0000 −478.076 2494.48 −471.778 −7505.29 −5766.62 6561.00 14528.9
1.4 14.1013 −81.0000 −313.152 −1990.91 −1142.21 5786.77 −11635.8 6561.00 −28074.4
1.5 20.6700 −81.0000 −84.7493 −1160.64 −1674.27 −6983.68 −12334.8 6561.00 −23990.6
1.6 43.1634 −81.0000 1351.08 −351.832 −3496.23 4650.87 36217.3 6561.00 −15186.2
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.b ✓ 6
3.b odd 2 1 171.10.a.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
57.10.a.b ✓ 6 1.a even 1 1 trivial
171.10.a.b 6 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T26−33T25−1410T24+41136T23+241968T22−9984384T2+43621632 T_{2}^{6} - 33T_{2}^{5} - 1410T_{2}^{4} + 41136T_{2}^{3} + 241968T_{2}^{2} - 9984384T_{2} + 43621632 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 T6−33T5+⋯+43621632 T^{6} - 33 T^{5} + \cdots + 43621632 Copy content Toggle raw display
33 (T+81)6 (T + 81)^{6} Copy content Toggle raw display
55 T6+⋯−17 ⁣⋯ ⁣00 T^{6} + \cdots - 17\!\cdots\!00 Copy content Toggle raw display
77 T6+⋯+47 ⁣⋯ ⁣68 T^{6} + \cdots + 47\!\cdots\!68 Copy content Toggle raw display
1111 T6+⋯−30 ⁣⋯ ⁣52 T^{6} + \cdots - 30\!\cdots\!52 Copy content Toggle raw display
1313 T6+⋯−65 ⁣⋯ ⁣00 T^{6} + \cdots - 65\!\cdots\!00 Copy content Toggle raw display
1717 T6+⋯+31 ⁣⋯ ⁣00 T^{6} + \cdots + 31\!\cdots\!00 Copy content Toggle raw display
1919 (T−130321)6 (T - 130321)^{6} Copy content Toggle raw display
2323 T6+⋯+20 ⁣⋯ ⁣76 T^{6} + \cdots + 20\!\cdots\!76 Copy content Toggle raw display
2929 T6+⋯−30 ⁣⋯ ⁣40 T^{6} + \cdots - 30\!\cdots\!40 Copy content Toggle raw display
3131 T6+⋯−17 ⁣⋯ ⁣88 T^{6} + \cdots - 17\!\cdots\!88 Copy content Toggle raw display
3737 T6+⋯−83 ⁣⋯ ⁣12 T^{6} + \cdots - 83\!\cdots\!12 Copy content Toggle raw display
4141 T6+⋯−22 ⁣⋯ ⁣84 T^{6} + \cdots - 22\!\cdots\!84 Copy content Toggle raw display
4343 T6+⋯+87 ⁣⋯ ⁣48 T^{6} + \cdots + 87\!\cdots\!48 Copy content Toggle raw display
4747 T6+⋯−12 ⁣⋯ ⁣48 T^{6} + \cdots - 12\!\cdots\!48 Copy content Toggle raw display
5353 T6+⋯−36 ⁣⋯ ⁣16 T^{6} + \cdots - 36\!\cdots\!16 Copy content Toggle raw display
5959 T6+⋯+32 ⁣⋯ ⁣56 T^{6} + \cdots + 32\!\cdots\!56 Copy content Toggle raw display
6161 T6+⋯−77 ⁣⋯ ⁣80 T^{6} + \cdots - 77\!\cdots\!80 Copy content Toggle raw display
6767 T6+⋯+98 ⁣⋯ ⁣60 T^{6} + \cdots + 98\!\cdots\!60 Copy content Toggle raw display
7171 T6+⋯−13 ⁣⋯ ⁣80 T^{6} + \cdots - 13\!\cdots\!80 Copy content Toggle raw display
7373 T6+⋯−15 ⁣⋯ ⁣08 T^{6} + \cdots - 15\!\cdots\!08 Copy content Toggle raw display
7979 T6+⋯+31 ⁣⋯ ⁣68 T^{6} + \cdots + 31\!\cdots\!68 Copy content Toggle raw display
8383 T6+⋯−19 ⁣⋯ ⁣08 T^{6} + \cdots - 19\!\cdots\!08 Copy content Toggle raw display
8989 T6+⋯+18 ⁣⋯ ⁣40 T^{6} + \cdots + 18\!\cdots\!40 Copy content Toggle raw display
9797 T6+⋯+19 ⁣⋯ ⁣28 T^{6} + \cdots + 19\!\cdots\!28 Copy content Toggle raw display
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