Properties

Label 483.4.a.i
Level 483483
Weight 44
Character orbit 483.a
Self dual yes
Analytic conductor 28.49828.498
Analytic rank 00
Dimension 99
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] = [483,4,Mod(1,483)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(483, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("483.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: N N == 483=3723 483 = 3 \cdot 7 \cdot 23
Weight: k k == 4 4
Character orbit: [χ][\chi] == 483.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: 28.497922532828.4979225328
Analytic rank: 00
Dimension: 99
Coefficient field: Q[x]/(x9)\mathbb{Q}[x]/(x^{9} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x9x851x7+34x6+861x5401x45403x3+1772x2+8716x192 x^{9} - x^{8} - 51x^{7} + 34x^{6} + 861x^{5} - 401x^{4} - 5403x^{3} + 1772x^{2} + 8716x - 192 Copy content Toggle raw display
Coefficient ring: Z[a1,,a5]\Z[a_1, \ldots, a_{5}]
Coefficient ring index: 2 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,,β81,\beta_1,\ldots,\beta_{8} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(β1+1)q2+3q3+(β22β1+4)q4+(β4+4)q5+(3β1+3)q6+7q7+(β3+2β2++15)q8+9q9+(β6β4+β2++9)q10++(9β7+9β4++45)q99+O(q100) q + ( - \beta_1 + 1) q^{2} + 3 q^{3} + (\beta_{2} - 2 \beta_1 + 4) q^{4} + ( - \beta_{4} + 4) q^{5} + ( - 3 \beta_1 + 3) q^{6} + 7 q^{7} + ( - \beta_{3} + 2 \beta_{2} + \cdots + 15) q^{8} + 9 q^{9} + ( - \beta_{6} - \beta_{4} + \beta_{2} + \cdots + 9) q^{10}+ \cdots + ( - 9 \beta_{7} + 9 \beta_{4} + \cdots + 45) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 9q+8q2+27q3+38q4+39q5+24q6+63q7+135q8+81q9+81q10+38q11+114q12+107q13+56q14+117q15+178q16+170q17+72q18++342q99+O(q100) 9 q + 8 q^{2} + 27 q^{3} + 38 q^{4} + 39 q^{5} + 24 q^{6} + 63 q^{7} + 135 q^{8} + 81 q^{9} + 81 q^{10} + 38 q^{11} + 114 q^{12} + 107 q^{13} + 56 q^{14} + 117 q^{15} + 178 q^{16} + 170 q^{17} + 72 q^{18}+ \cdots + 342 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x9x851x7+34x6+861x5401x45403x3+1772x2+8716x192 x^{9} - x^{8} - 51x^{7} + 34x^{6} + 861x^{5} - 401x^{4} - 5403x^{3} + 1772x^{2} + 8716x - 192 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν211 \nu^{2} - 11 Copy content Toggle raw display
β3\beta_{3}== ν3ν217ν+8 \nu^{3} - \nu^{2} - 17\nu + 8 Copy content Toggle raw display
β4\beta_{4}== (18ν8+57ν7+426ν61535ν5+2445ν4+8922ν373899ν2++118608)/5896 ( - 18 \nu^{8} + 57 \nu^{7} + 426 \nu^{6} - 1535 \nu^{5} + 2445 \nu^{4} + 8922 \nu^{3} - 73899 \nu^{2} + \cdots + 118608 ) / 5896 Copy content Toggle raw display
β5\beta_{5}== (35ν8+12ν71811ν6905ν5+28288ν4+18519ν3135262ν2++79896)/5896 ( 35 \nu^{8} + 12 \nu^{7} - 1811 \nu^{6} - 905 \nu^{5} + 28288 \nu^{4} + 18519 \nu^{3} - 135262 \nu^{2} + \cdots + 79896 ) / 5896 Copy content Toggle raw display
β6\beta_{6}== (39ν8+492ν7+923ν617943ν51704ν4+171153ν3+31920)/5896 ( - 39 \nu^{8} + 492 \nu^{7} + 923 \nu^{6} - 17943 \nu^{5} - 1704 \nu^{4} + 171153 \nu^{3} + \cdots - 31920 ) / 5896 Copy content Toggle raw display
β7\beta_{7}== (8ν83ν7368ν625ν5+5457ν4+948ν329131ν2+1974ν+30544)/536 ( 8\nu^{8} - 3\nu^{7} - 368\nu^{6} - 25\nu^{5} + 5457\nu^{4} + 948\nu^{3} - 29131\nu^{2} + 1974\nu + 30544 ) / 536 Copy content Toggle raw display
β8\beta_{8}== (115ν8487ν74687ν6+18610ν5+57465ν4207841ν3++175760)/5896 ( 115 \nu^{8} - 487 \nu^{7} - 4687 \nu^{6} + 18610 \nu^{5} + 57465 \nu^{4} - 207841 \nu^{3} + \cdots + 175760 ) / 5896 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β2+11 \beta_{2} + 11 Copy content Toggle raw display
ν3\nu^{3}== β3+β2+17β1+3 \beta_{3} + \beta_{2} + 17\beta _1 + 3 Copy content Toggle raw display
ν4\nu^{4}== β72β5+β4+3β3+24β2+8β1+190 \beta_{7} - 2\beta_{5} + \beta_{4} + 3\beta_{3} + 24\beta_{2} + 8\beta _1 + 190 Copy content Toggle raw display
ν5\nu^{5}== 3β8+3β66β5+β4+36β3+42β2+336β1+162 3\beta_{8} + 3\beta_{6} - 6\beta_{5} + \beta_{4} + 36\beta_{3} + 42\beta_{2} + 336\beta _1 + 162 Copy content Toggle raw display
ν6\nu^{6}== 3β8+36β7+5β684β5+21β4+129β3+568β2+407β1+3818 3\beta_{8} + 36\beta_{7} + 5\beta_{6} - 84\beta_{5} + 21\beta_{4} + 129\beta_{3} + 568\beta_{2} + 407\beta _1 + 3818 Copy content Toggle raw display
ν7\nu^{7}== 119β8+19β7+135β6276β5+24β4+1073β3++6141 119 \beta_{8} + 19 \beta_{7} + 135 \beta_{6} - 276 \beta_{5} + 24 \beta_{4} + 1073 \beta_{3} + \cdots + 6141 Copy content Toggle raw display
ν8\nu^{8}== 192β8+1048β7+290β62622β5+296β4+4284β3++84715 192 \beta_{8} + 1048 \beta_{7} + 290 \beta_{6} - 2622 \beta_{5} + 296 \beta_{4} + 4284 \beta_{3} + \cdots + 84715 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
5.24907
3.86754
3.08695
1.79037
0.0219372
−1.35308
−3.40804
−3.78226
−4.47249
−4.24907 3.00000 10.0546 3.00025 −12.7472 7.00000 −8.73012 9.00000 −12.7483
1.2 −2.86754 3.00000 0.222786 −10.1020 −8.60262 7.00000 22.3015 9.00000 28.9679
1.3 −2.08695 3.00000 −3.64464 21.1286 −6.26085 7.00000 24.3018 9.00000 −44.0944
1.4 −0.790374 3.00000 −7.37531 7.57526 −2.37112 7.00000 12.1522 9.00000 −5.98729
1.5 0.978063 3.00000 −7.04339 −16.1805 2.93419 7.00000 −14.7134 9.00000 −15.8256
1.6 2.35308 3.00000 −2.46299 11.9881 7.05925 7.00000 −24.6203 9.00000 28.2090
1.7 4.40804 3.00000 11.4308 18.5472 13.2241 7.00000 15.1232 9.00000 81.7570
1.8 4.78226 3.00000 14.8700 −5.89437 14.3468 7.00000 32.8541 9.00000 −28.1884
1.9 5.47249 3.00000 21.9481 8.93744 16.4175 7.00000 76.3311 9.00000 48.9100
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

p p Sign
33 1 -1
77 1 -1
2323 +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 483.4.a.i 9
3.b odd 2 1 1449.4.a.j 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
483.4.a.i 9 1.a even 1 1 trivial
1449.4.a.j 9 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T298T2823T27+267T26+64T252685T24+526T23+8786T221400T25336 T_{2}^{9} - 8T_{2}^{8} - 23T_{2}^{7} + 267T_{2}^{6} + 64T_{2}^{5} - 2685T_{2}^{4} + 526T_{2}^{3} + 8786T_{2}^{2} - 1400T_{2} - 5336 acting on S4new(Γ0(483))S_{4}^{\mathrm{new}}(\Gamma_0(483)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T98T8+5336 T^{9} - 8 T^{8} + \cdots - 5336 Copy content Toggle raw display
33 (T3)9 (T - 3)^{9} Copy content Toggle raw display
55 T939T8++919405312 T^{9} - 39 T^{8} + \cdots + 919405312 Copy content Toggle raw display
77 (T7)9 (T - 7)^{9} Copy content Toggle raw display
1111 T9+4877431083904 T^{9} + \cdots - 4877431083904 Copy content Toggle raw display
1313 T9+11393014127088 T^{9} + \cdots - 11393014127088 Copy content Toggle raw display
1717 T9+11 ⁣ ⁣56 T^{9} + \cdots - 11\!\cdots\!56 Copy content Toggle raw display
1919 T9+31 ⁣ ⁣12 T^{9} + \cdots - 31\!\cdots\!12 Copy content Toggle raw display
2323 (T+23)9 (T + 23)^{9} Copy content Toggle raw display
2929 T9++59 ⁣ ⁣04 T^{9} + \cdots + 59\!\cdots\!04 Copy content Toggle raw display
3131 T9++11 ⁣ ⁣12 T^{9} + \cdots + 11\!\cdots\!12 Copy content Toggle raw display
3737 T9+32 ⁣ ⁣68 T^{9} + \cdots - 32\!\cdots\!68 Copy content Toggle raw display
4141 T9++17 ⁣ ⁣24 T^{9} + \cdots + 17\!\cdots\!24 Copy content Toggle raw display
4343 T9++21 ⁣ ⁣36 T^{9} + \cdots + 21\!\cdots\!36 Copy content Toggle raw display
4747 T9++51 ⁣ ⁣08 T^{9} + \cdots + 51\!\cdots\!08 Copy content Toggle raw display
5353 T9++81 ⁣ ⁣24 T^{9} + \cdots + 81\!\cdots\!24 Copy content Toggle raw display
5959 T9+11 ⁣ ⁣36 T^{9} + \cdots - 11\!\cdots\!36 Copy content Toggle raw display
6161 T9+14 ⁣ ⁣08 T^{9} + \cdots - 14\!\cdots\!08 Copy content Toggle raw display
6767 T9+44 ⁣ ⁣68 T^{9} + \cdots - 44\!\cdots\!68 Copy content Toggle raw display
7171 T9+61 ⁣ ⁣44 T^{9} + \cdots - 61\!\cdots\!44 Copy content Toggle raw display
7373 T9+91 ⁣ ⁣16 T^{9} + \cdots - 91\!\cdots\!16 Copy content Toggle raw display
7979 T9++22 ⁣ ⁣72 T^{9} + \cdots + 22\!\cdots\!72 Copy content Toggle raw display
8383 T9+21 ⁣ ⁣68 T^{9} + \cdots - 21\!\cdots\!68 Copy content Toggle raw display
8989 T9+11 ⁣ ⁣32 T^{9} + \cdots - 11\!\cdots\!32 Copy content Toggle raw display
9797 T9+24 ⁣ ⁣88 T^{9} + \cdots - 24\!\cdots\!88 Copy content Toggle raw display
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