Properties

Label 152.8.a.d
Level 152152
Weight 88
Character orbit 152.a
Self dual yes
Analytic conductor 47.48347.483
Analytic rank 00
Dimension 99
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] = [152,8,Mod(1,152)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(152, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("152.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Level: N N == 152=2319 152 = 2^{3} \cdot 19
Weight: k k == 8 8
Character orbit: [χ][\chi] == 152.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,0,56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 47.482523873647.4825238736
Analytic rank: 00
Dimension: 99
Coefficient field: Q[x]/(x9)\mathbb{Q}[x]/(x^{9} - \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x92x813922x725112x6+57411673x5+379057666x462486804160x3++69542466153984 x^{9} - 2 x^{8} - 13922 x^{7} - 25112 x^{6} + 57411673 x^{5} + 379057666 x^{4} - 62486804160 x^{3} + \cdots + 69542466153984 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 21935 2^{19}\cdot 3\cdot 5
Twist minimal: yes
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,,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+6)q3+(β22β1+10)q5+(β3+β2+52)q7+(β6+β3+3β2++943)q9+(β7β5β4++375)q11++(336β8+829β7++2420371)q99+O(q100) q + (\beta_1 + 6) q^{3} + (\beta_{2} - 2 \beta_1 + 10) q^{5} + (\beta_{3} + \beta_{2} + 52) q^{7} + (\beta_{6} + \beta_{3} + 3 \beta_{2} + \cdots + 943) q^{9} + ( - \beta_{7} - \beta_{5} - \beta_{4} + \cdots + 375) q^{11}+ \cdots + (336 \beta_{8} + 829 \beta_{7} + \cdots + 2420371) q^{99}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 9q+56q3+82q5+464q7+8513q9+3360q11+16522q1346608q1512902q17+61731q19+956q2176956q23+188627q25+417356q27+63186q29+314040q31++21789816q99+O(q100) 9 q + 56 q^{3} + 82 q^{5} + 464 q^{7} + 8513 q^{9} + 3360 q^{11} + 16522 q^{13} - 46608 q^{15} - 12902 q^{17} + 61731 q^{19} + 956 q^{21} - 76956 q^{23} + 188627 q^{25} + 417356 q^{27} + 63186 q^{29} + 314040 q^{31}+ \cdots + 21789816 q^{99}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x92x813922x725112x6+57411673x5+379057666x462486804160x3++69542466153984 x^{9} - 2 x^{8} - 13922 x^{7} - 25112 x^{6} + 57411673 x^{5} + 379057666 x^{4} - 62486804160 x^{3} + \cdots + 69542466153984 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (86369939727737ν8++12 ⁣ ⁣44)/12 ⁣ ⁣96 ( 86369939727737 \nu^{8} + \cdots + 12\!\cdots\!44 ) / 12\!\cdots\!96 Copy content Toggle raw display
β3\beta_{3}== (14 ⁣ ⁣55ν8++20 ⁣ ⁣12)/19 ⁣ ⁣36 ( 14\!\cdots\!55 \nu^{8} + \cdots + 20\!\cdots\!12 ) / 19\!\cdots\!36 Copy content Toggle raw display
β4\beta_{4}== (67 ⁣ ⁣97ν8++58 ⁣ ⁣68)/29 ⁣ ⁣04 ( - 67\!\cdots\!97 \nu^{8} + \cdots + 58\!\cdots\!68 ) / 29\!\cdots\!04 Copy content Toggle raw display
β5\beta_{5}== (424739328970720ν8+47 ⁣ ⁣00)/18 ⁣ ⁣44 ( - 424739328970720 \nu^{8} + \cdots - 47\!\cdots\!00 ) / 18\!\cdots\!44 Copy content Toggle raw display
β6\beta_{6}== (55 ⁣ ⁣31ν8+87 ⁣ ⁣08)/19 ⁣ ⁣36 ( - 55\!\cdots\!31 \nu^{8} + \cdots - 87\!\cdots\!08 ) / 19\!\cdots\!36 Copy content Toggle raw display
β7\beta_{7}== (772928886362595ν8+75 ⁣ ⁣56)/12 ⁣ ⁣96 ( - 772928886362595 \nu^{8} + \cdots - 75\!\cdots\!56 ) / 12\!\cdots\!96 Copy content Toggle raw display
β8\beta_{8}== (84 ⁣ ⁣43ν8+41 ⁣ ⁣64)/48 ⁣ ⁣84 ( - 84\!\cdots\!43 \nu^{8} + \cdots - 41\!\cdots\!64 ) / 48\!\cdots\!84 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β6+β3+3β2+5β1+3094 \beta_{6} + \beta_{3} + 3\beta_{2} + 5\beta _1 + 3094 Copy content Toggle raw display
ν3\nu^{3}== 3β86β73β6+14β522β4+5β3170β2+5737β1+16306 -3\beta_{8} - 6\beta_{7} - 3\beta_{6} + 14\beta_{5} - 22\beta_{4} + 5\beta_{3} - 170\beta_{2} + 5737\beta _1 + 16306 Copy content Toggle raw display
ν4\nu^{4}== 246β8870β7+6745β6+218β5+164β4+7065β3++17599490 246 \beta_{8} - 870 \beta_{7} + 6745 \beta_{6} + 218 \beta_{5} + 164 \beta_{4} + 7065 \beta_{3} + \cdots + 17599490 Copy content Toggle raw display
ν5\nu^{5}== 30519β822602β718885β6+135952β5156332β4++126800870 - 30519 \beta_{8} - 22602 \beta_{7} - 18885 \beta_{6} + 135952 \beta_{5} - 156332 \beta_{4} + \cdots + 126800870 Copy content Toggle raw display
ν6\nu^{6}== 3257124β88998014β7+43235341β6+2992102β5+938212β4++109685571738 3257124 \beta_{8} - 8998014 \beta_{7} + 43235341 \beta_{6} + 2992102 \beta_{5} + 938212 \beta_{4} + \cdots + 109685571738 Copy content Toggle raw display
ν7\nu^{7}== 223599417β815333714β7157559025β6+1091696124β5++937381890942 - 223599417 \beta_{8} - 15333714 \beta_{7} - 157559025 \beta_{6} + 1091696124 \beta_{5} + \cdots + 937381890942 Copy content Toggle raw display
ν8\nu^{8}== 31102975458β874462823846β7+277014842209β6+28990339890β5++702834772848850 31102975458 \beta_{8} - 74462823846 \beta_{7} + 277014842209 \beta_{6} + 28990339890 \beta_{5} + \cdots + 702834772848850 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.

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
−82.7282
−70.2165
−30.7875
−18.9403
−7.53304
10.1837
41.7024
77.2992
83.0202
0 −76.7282 0 510.168 0 −402.237 0 3700.22 0
1.2 0 −64.2165 0 −1.14422 0 1813.97 0 1936.76 0
1.3 0 −24.7875 0 150.360 0 −1661.39 0 −1572.58 0
1.4 0 −12.9403 0 −448.356 0 −610.684 0 −2019.55 0
1.5 0 −1.53304 0 −60.7705 0 1098.49 0 −2184.65 0
1.6 0 16.1837 0 −161.818 0 −301.718 0 −1925.09 0
1.7 0 47.7024 0 432.335 0 207.442 0 88.5147 0
1.8 0 83.2992 0 −427.913 0 −991.087 0 4751.76 0
1.9 0 89.0202 0 89.1384 0 1311.21 0 5737.60 0
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
22 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 152.8.a.d 9
4.b odd 2 1 304.8.a.m 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
152.8.a.d 9 1.a even 1 1 trivial
304.8.a.m 9 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T3956T3812530T37+539452T36+47978161T35++13870109203776 T_{3}^{9} - 56 T_{3}^{8} - 12530 T_{3}^{7} + 539452 T_{3}^{6} + 47978161 T_{3}^{5} + \cdots + 13870109203776 acting on S8new(Γ0(152))S_{8}^{\mathrm{new}}(\Gamma_0(152)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T9 T^{9} Copy content Toggle raw display
33 T9++13870109203776 T^{9} + \cdots + 13870109203776 Copy content Toggle raw display
55 T9++63 ⁣ ⁣00 T^{9} + \cdots + 63\!\cdots\!00 Copy content Toggle raw display
77 T9++66 ⁣ ⁣00 T^{9} + \cdots + 66\!\cdots\!00 Copy content Toggle raw display
1111 T9++58 ⁣ ⁣00 T^{9} + \cdots + 58\!\cdots\!00 Copy content Toggle raw display
1313 T9++12 ⁣ ⁣12 T^{9} + \cdots + 12\!\cdots\!12 Copy content Toggle raw display
1717 T9+77 ⁣ ⁣30 T^{9} + \cdots - 77\!\cdots\!30 Copy content Toggle raw display
1919 (T6859)9 (T - 6859)^{9} Copy content Toggle raw display
2323 T9++75 ⁣ ⁣84 T^{9} + \cdots + 75\!\cdots\!84 Copy content Toggle raw display
2929 T9++10 ⁣ ⁣12 T^{9} + \cdots + 10\!\cdots\!12 Copy content Toggle raw display
3131 T9++48 ⁣ ⁣56 T^{9} + \cdots + 48\!\cdots\!56 Copy content Toggle raw display
3737 T9++72 ⁣ ⁣12 T^{9} + \cdots + 72\!\cdots\!12 Copy content Toggle raw display
4141 T9+54 ⁣ ⁣84 T^{9} + \cdots - 54\!\cdots\!84 Copy content Toggle raw display
4343 T9++11 ⁣ ⁣52 T^{9} + \cdots + 11\!\cdots\!52 Copy content Toggle raw display
4747 T9+11 ⁣ ⁣08 T^{9} + \cdots - 11\!\cdots\!08 Copy content Toggle raw display
5353 T9+72 ⁣ ⁣24 T^{9} + \cdots - 72\!\cdots\!24 Copy content Toggle raw display
5959 T9++71 ⁣ ⁣44 T^{9} + \cdots + 71\!\cdots\!44 Copy content Toggle raw display
6161 T9++18 ⁣ ⁣00 T^{9} + \cdots + 18\!\cdots\!00 Copy content Toggle raw display
6767 T9++50 ⁣ ⁣68 T^{9} + \cdots + 50\!\cdots\!68 Copy content Toggle raw display
7171 T9+10 ⁣ ⁣20 T^{9} + \cdots - 10\!\cdots\!20 Copy content Toggle raw display
7373 T9+30 ⁣ ⁣34 T^{9} + \cdots - 30\!\cdots\!34 Copy content Toggle raw display
7979 T9+61 ⁣ ⁣60 T^{9} + \cdots - 61\!\cdots\!60 Copy content Toggle raw display
8383 T9++29 ⁣ ⁣40 T^{9} + \cdots + 29\!\cdots\!40 Copy content Toggle raw display
8989 T9+37 ⁣ ⁣72 T^{9} + \cdots - 37\!\cdots\!72 Copy content Toggle raw display
9797 T9++39 ⁣ ⁣44 T^{9} + \cdots + 39\!\cdots\!44 Copy content Toggle raw display
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