Properties

Label 585.6.a.p
Level 585585
Weight 66
Character orbit 585.a
Self dual yes
Analytic conductor 93.82593.825
Analytic rank 00
Dimension 99
CM no
Inner twists 11

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [585,6,Mod(1,585)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(585, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("585.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Level: N N == 585=32513 585 = 3^{2} \cdot 5 \cdot 13
Weight: k k == 6 6
Character orbit: [χ][\chi] == 585.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,5,0,139] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: 93.824534590693.8245345906
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: x94x8205x7+608x6+13727x527536x4346839x3+433844x2+3899136 x^{9} - 4 x^{8} - 205 x^{7} + 608 x^{6} + 13727 x^{5} - 27536 x^{4} - 346839 x^{3} + 433844 x^{2} + \cdots - 3899136 Copy content Toggle raw display
Coefficient ring: Z[a1,,a11]\Z[a_1, \ldots, a_{11}]
Coefficient ring index: 2632 2^{6}\cdot 3^{2}
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+1)q2+(β2β1+16)q425q5+(β6β23β116)q7+(β6β512β1+33)q8+(25β125)q10+(β8β7β6++53)q11++(44β8+18β7++12468)q98+O(q100) q + ( - \beta_1 + 1) q^{2} + (\beta_{2} - \beta_1 + 16) q^{4} - 25 q^{5} + (\beta_{6} - \beta_{2} - 3 \beta_1 - 16) q^{7} + (\beta_{6} - \beta_{5} - 12 \beta_1 + 33) q^{8} + (25 \beta_1 - 25) q^{10} + (\beta_{8} - \beta_{7} - \beta_{6} + \cdots + 53) q^{11}+ \cdots + (44 \beta_{8} + 18 \beta_{7} + \cdots + 12468) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 9q+5q2+139q4225q5153q7+255q8125q10+503q11+1521q13+1110q14+763q16+1897q172412q193475q203776q22+3901q23+5625q25++119225q98+O(q100) 9 q + 5 q^{2} + 139 q^{4} - 225 q^{5} - 153 q^{7} + 255 q^{8} - 125 q^{10} + 503 q^{11} + 1521 q^{13} + 1110 q^{14} + 763 q^{16} + 1897 q^{17} - 2412 q^{19} - 3475 q^{20} - 3776 q^{22} + 3901 q^{23} + 5625 q^{25}+ \cdots + 119225 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x94x8205x7+608x6+13727x527536x4346839x3+433844x2+3899136 x^{9} - 4 x^{8} - 205 x^{7} + 608 x^{6} + 13727 x^{5} - 27536 x^{4} - 346839 x^{3} + 433844 x^{2} + \cdots - 3899136 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== ν2ν47 \nu^{2} - \nu - 47 Copy content Toggle raw display
β3\beta_{3}== (59ν8+351ν7+10912ν655376ν5601957ν4+2476749ν3++113984)/45280 ( - 59 \nu^{8} + 351 \nu^{7} + 10912 \nu^{6} - 55376 \nu^{5} - 601957 \nu^{4} + 2476749 \nu^{3} + \cdots + 113984 ) / 45280 Copy content Toggle raw display
β4\beta_{4}== (587ν82859ν7109064ν6+410960ν5+6097941ν415477745ν3++182771520)/271680 ( 587 \nu^{8} - 2859 \nu^{7} - 109064 \nu^{6} + 410960 \nu^{5} + 6097941 \nu^{4} - 15477745 \nu^{3} + \cdots + 182771520 ) / 271680 Copy content Toggle raw display
β5\beta_{5}== (109ν81013ν717128ν6+162320ν5+669267ν47383375ν3+132330560)/90560 ( 109 \nu^{8} - 1013 \nu^{7} - 17128 \nu^{6} + 162320 \nu^{5} + 669267 \nu^{4} - 7383375 \nu^{3} + \cdots - 132330560 ) / 90560 Copy content Toggle raw display
β6\beta_{6}== (109ν81013ν717128ν6+162320ν5+669267ν47473935ν3+141024320)/90560 ( 109 \nu^{8} - 1013 \nu^{7} - 17128 \nu^{6} + 162320 \nu^{5} + 669267 \nu^{4} - 7473935 \nu^{3} + \cdots - 141024320 ) / 90560 Copy content Toggle raw display
β7\beta_{7}== (277ν8+1245ν7+53092ν6173992ν53185691ν4+6123863ν3+246852672)/135840 ( - 277 \nu^{8} + 1245 \nu^{7} + 53092 \nu^{6} - 173992 \nu^{5} - 3185691 \nu^{4} + 6123863 \nu^{3} + \cdots - 246852672 ) / 135840 Copy content Toggle raw display
β8\beta_{8}== (185ν8+1657ν7+31376ν6261664ν51521415ν4+11355611ν3++109852736)/90560 ( - 185 \nu^{8} + 1657 \nu^{7} + 31376 \nu^{6} - 261664 \nu^{5} - 1521415 \nu^{4} + 11355611 \nu^{3} + \cdots + 109852736 ) / 90560 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β2+β1+47 \beta_{2} + \beta _1 + 47 Copy content Toggle raw display
ν3\nu^{3}== β6+β5+3β2+76β1+45 -\beta_{6} + \beta_{5} + 3\beta_{2} + 76\beta _1 + 45 Copy content Toggle raw display
ν4\nu^{4}== β83β72β6+4β53β4+117β2+172β1+3583 \beta_{8} - 3\beta_{7} - 2\beta_{6} + 4\beta_{5} - 3\beta_{4} + 117\beta_{2} + 172\beta _1 + 3583 Copy content Toggle raw display
ν5\nu^{5}== 9β811β7143β6+155β517β414β3+503β2++8013 9 \beta_{8} - 11 \beta_{7} - 143 \beta_{6} + 155 \beta_{5} - 17 \beta_{4} - 14 \beta_{3} + 503 \beta_{2} + \cdots + 8013 Copy content Toggle raw display
ν6\nu^{6}== 206β8498β7450β6+850β5486β4+20β3++328425 206 \beta_{8} - 498 \beta_{7} - 450 \beta_{6} + 850 \beta_{5} - 486 \beta_{4} + 20 \beta_{3} + \cdots + 328425 Copy content Toggle raw display
ν7\nu^{7}== 1980β82308β717153β6+19777β53196β42368β3++1138793 1980 \beta_{8} - 2308 \beta_{7} - 17153 \beta_{6} + 19777 \beta_{5} - 3196 \beta_{4} - 2368 \beta_{3} + \cdots + 1138793 Copy content Toggle raw display
ν8\nu^{8}== 31229β864903β772630β6+130552β562335β4++33126463 31229 \beta_{8} - 64903 \beta_{7} - 72630 \beta_{6} + 130552 \beta_{5} - 62335 \beta_{4} + \cdots + 33126463 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
10.9498
7.93845
6.88132
2.31435
1.92674
−4.49910
−4.69326
−7.19157
−9.62674
−9.94981 0 66.9988 −25.0000 0 −113.847 −348.231 0 248.745
1.2 −6.93845 0 16.1421 −25.0000 0 −138.927 110.029 0 173.461
1.3 −5.88132 0 2.58998 −25.0000 0 111.916 172.970 0 147.033
1.4 −1.31435 0 −30.2725 −25.0000 0 80.6033 81.8477 0 32.8587
1.5 −0.926735 0 −31.1412 −25.0000 0 −27.0415 58.5151 0 23.1684
1.6 5.49910 0 −1.75989 −25.0000 0 −180.806 −185.649 0 −137.478
1.7 5.69326 0 0.413187 −25.0000 0 161.755 −179.832 0 −142.331
1.8 8.19157 0 35.1018 −25.0000 0 −132.038 25.4090 0 −204.789
1.9 10.6267 0 80.9276 −25.0000 0 85.3847 519.941 0 −265.669
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
55 +1 +1
1313 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 585.6.a.p yes 9
3.b odd 2 1 585.6.a.o 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.6.a.o 9 3.b odd 2 1
585.6.a.p yes 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T295T28201T27+855T26+12972T2542890T24++1347840 T_{2}^{9} - 5 T_{2}^{8} - 201 T_{2}^{7} + 855 T_{2}^{6} + 12972 T_{2}^{5} - 42890 T_{2}^{4} + \cdots + 1347840 acting on S6new(Γ0(585))S_{6}^{\mathrm{new}}(\Gamma_0(585)). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T95T8++1347840 T^{9} - 5 T^{8} + \cdots + 1347840 Copy content Toggle raw display
33 T9 T^{9} Copy content Toggle raw display
55 (T+25)9 (T + 25)^{9} Copy content Toggle raw display
77 T9++12 ⁣ ⁣04 T^{9} + \cdots + 12\!\cdots\!04 Copy content Toggle raw display
1111 T9++32 ⁣ ⁣76 T^{9} + \cdots + 32\!\cdots\!76 Copy content Toggle raw display
1313 (T169)9 (T - 169)^{9} Copy content Toggle raw display
1717 T9+23 ⁣ ⁣44 T^{9} + \cdots - 23\!\cdots\!44 Copy content Toggle raw display
1919 T9+22 ⁣ ⁣64 T^{9} + \cdots - 22\!\cdots\!64 Copy content Toggle raw display
2323 T9+10 ⁣ ⁣32 T^{9} + \cdots - 10\!\cdots\!32 Copy content Toggle raw display
2929 T9+58 ⁣ ⁣00 T^{9} + \cdots - 58\!\cdots\!00 Copy content Toggle raw display
3131 T9++36 ⁣ ⁣72 T^{9} + \cdots + 36\!\cdots\!72 Copy content Toggle raw display
3737 T9+11 ⁣ ⁣04 T^{9} + \cdots - 11\!\cdots\!04 Copy content Toggle raw display
4141 T9++79 ⁣ ⁣36 T^{9} + \cdots + 79\!\cdots\!36 Copy content Toggle raw display
4343 T9+99 ⁣ ⁣12 T^{9} + \cdots - 99\!\cdots\!12 Copy content Toggle raw display
4747 T9++18 ⁣ ⁣28 T^{9} + \cdots + 18\!\cdots\!28 Copy content Toggle raw display
5353 T9++29 ⁣ ⁣00 T^{9} + \cdots + 29\!\cdots\!00 Copy content Toggle raw display
5959 T9++78 ⁣ ⁣84 T^{9} + \cdots + 78\!\cdots\!84 Copy content Toggle raw display
6161 T9++22 ⁣ ⁣00 T^{9} + \cdots + 22\!\cdots\!00 Copy content Toggle raw display
6767 T9++20 ⁣ ⁣00 T^{9} + \cdots + 20\!\cdots\!00 Copy content Toggle raw display
7171 T9++60 ⁣ ⁣48 T^{9} + \cdots + 60\!\cdots\!48 Copy content Toggle raw display
7373 T9+69 ⁣ ⁣00 T^{9} + \cdots - 69\!\cdots\!00 Copy content Toggle raw display
7979 T9++73 ⁣ ⁣00 T^{9} + \cdots + 73\!\cdots\!00 Copy content Toggle raw display
8383 T9+41 ⁣ ⁣60 T^{9} + \cdots - 41\!\cdots\!60 Copy content Toggle raw display
8989 T9+33 ⁣ ⁣60 T^{9} + \cdots - 33\!\cdots\!60 Copy content Toggle raw display
9797 T9+73 ⁣ ⁣00 T^{9} + \cdots - 73\!\cdots\!00 Copy content Toggle raw display
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