Properties

Label 504.6.s.f
Level 504504
Weight 66
Character orbit 504.s
Analytic conductor 80.83380.833
Analytic rank 00
Dimension 1212
Inner twists 22

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [504,6,Mod(289,504)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(504, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 0, 2])) 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("504.289"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Level: N N == 504=23327 504 = 2^{3} \cdot 3^{2} \cdot 7
Weight: k k == 6 6
Character orbit: [χ][\chi] == 504.s (of order 33, degree 22, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,0,-17] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 80.833445185780.8334451857
Analytic rank: 00
Dimension: 1212
Relative dimension: 66 over Q(ζ3)\Q(\zeta_{3})
Coefficient field: Q[x]/(x12)\mathbb{Q}[x]/(x^{12} - \cdots)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x12x11+2041x1063452x9+3932036x870117724x7+1560078988x6++472919482810944 x^{12} - x^{11} + 2041 x^{10} - 63452 x^{9} + 3932036 x^{8} - 70117724 x^{7} + 1560078988 x^{6} + \cdots + 472919482810944 Copy content Toggle raw display
Coefficient ring: Z[a1,,a17]\Z[a_1, \ldots, a_{17}]
Coefficient ring index: 218374 2^{18}\cdot 3\cdot 7^{4}
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

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,,β111,\beta_1,\ldots,\beta_{11} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(β4β23β1)q5+(β6+9β1+8)q7+(β11+β6β5++94)q11+(β10β9+β6+72)q13++(98β11+93β10++2190)q97+O(q100) q + (\beta_{4} - \beta_{2} - 3 \beta_1) q^{5} + ( - \beta_{6} + 9 \beta_1 + 8) q^{7} + ( - \beta_{11} + \beta_{6} - \beta_{5} + \cdots + 94) q^{11} + (\beta_{10} - \beta_{9} + \beta_{6} + \cdots - 72) q^{13}+ \cdots + ( - 98 \beta_{11} + 93 \beta_{10} + \cdots + 2190) q^{97}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 12q17q5+144q7+565q11842q1352q17+107q19700q234881q2512958q293552q31+938q353453q37+15148q41+38278q4312136q47++28934q97+O(q100) 12 q - 17 q^{5} + 144 q^{7} + 565 q^{11} - 842 q^{13} - 52 q^{17} + 107 q^{19} - 700 q^{23} - 4881 q^{25} - 12958 q^{29} - 3552 q^{31} + 938 q^{35} - 3453 q^{37} + 15148 q^{41} + 38278 q^{43} - 12136 q^{47}+ \cdots + 28934 q^{97}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x12x11+2041x1063452x9+3932036x870117724x7+1560078988x6++472919482810944 x^{12} - x^{11} + 2041 x^{10} - 63452 x^{9} + 3932036 x^{8} - 70117724 x^{7} + 1560078988 x^{6} + \cdots + 472919482810944 : Copy content Toggle raw display

β1\beta_{1}== (84 ⁣ ⁣10ν11++67 ⁣ ⁣64)/88 ⁣ ⁣12 ( 84\!\cdots\!10 \nu^{11} + \cdots + 67\!\cdots\!64 ) / 88\!\cdots\!12 Copy content Toggle raw display
β2\beta_{2}== (54 ⁣ ⁣41ν11++13 ⁣ ⁣16)/92 ⁣ ⁣12 ( 54\!\cdots\!41 \nu^{11} + \cdots + 13\!\cdots\!16 ) / 92\!\cdots\!12 Copy content Toggle raw display
β3\beta_{3}== (10 ⁣ ⁣85ν11++27 ⁣ ⁣88)/13 ⁣ ⁣60 ( 10\!\cdots\!85 \nu^{11} + \cdots + 27\!\cdots\!88 ) / 13\!\cdots\!60 Copy content Toggle raw display
β4\beta_{4}== (23 ⁣ ⁣15ν11++93 ⁣ ⁣40)/22 ⁣ ⁣76 ( - 23\!\cdots\!15 \nu^{11} + \cdots + 93\!\cdots\!40 ) / 22\!\cdots\!76 Copy content Toggle raw display
β5\beta_{5}== (21 ⁣ ⁣01ν11+15 ⁣ ⁣96)/13 ⁣ ⁣60 ( 21\!\cdots\!01 \nu^{11} + \cdots - 15\!\cdots\!96 ) / 13\!\cdots\!60 Copy content Toggle raw display
β6\beta_{6}== (10 ⁣ ⁣98ν11+14 ⁣ ⁣96)/68 ⁣ ⁣80 ( - 10\!\cdots\!98 \nu^{11} + \cdots - 14\!\cdots\!96 ) / 68\!\cdots\!80 Copy content Toggle raw display
β7\beta_{7}== (86 ⁣ ⁣95ν11+24 ⁣ ⁣52)/45 ⁣ ⁣20 ( 86\!\cdots\!95 \nu^{11} + \cdots - 24\!\cdots\!52 ) / 45\!\cdots\!20 Copy content Toggle raw display
β8\beta_{8}== (33 ⁣ ⁣43ν11++58 ⁣ ⁣68)/13 ⁣ ⁣60 ( 33\!\cdots\!43 \nu^{11} + \cdots + 58\!\cdots\!68 ) / 13\!\cdots\!60 Copy content Toggle raw display
β9\beta_{9}== (20 ⁣ ⁣93ν11+28 ⁣ ⁣48)/32 ⁣ ⁣80 ( - 20\!\cdots\!93 \nu^{11} + \cdots - 28\!\cdots\!48 ) / 32\!\cdots\!80 Copy content Toggle raw display
β10\beta_{10}== (24 ⁣ ⁣87ν11+25 ⁣ ⁣92)/34 ⁣ ⁣40 ( - 24\!\cdots\!87 \nu^{11} + \cdots - 25\!\cdots\!92 ) / 34\!\cdots\!40 Copy content Toggle raw display
β11\beta_{11}== (15 ⁣ ⁣31ν11+90 ⁣ ⁣56)/19 ⁣ ⁣80 ( 15\!\cdots\!31 \nu^{11} + \cdots - 90\!\cdots\!56 ) / 19\!\cdots\!80 Copy content Toggle raw display
ν\nu== (β11+β10+β73β63β511β4+β3+2β29β1+10)/56 ( -\beta_{11} + \beta_{10} + \beta_{7} - 3\beta_{6} - 3\beta_{5} - 11\beta_{4} + \beta_{3} + 2\beta_{2} - 9\beta _1 + 10 ) / 56 Copy content Toggle raw display
ν2\nu^{2}== (12β117β10+16β9+27β816β7248β6+107β5++88)/56 ( 12 \beta_{11} - 7 \beta_{10} + 16 \beta_{9} + 27 \beta_{8} - 16 \beta_{7} - 248 \beta_{6} + 107 \beta_{5} + \cdots + 88 ) / 56 Copy content Toggle raw display
ν3\nu^{3}== (881β1180β101767β92949β8+11189β6+447β5++854552)/56 ( 881 \beta_{11} - 80 \beta_{10} - 1767 \beta_{9} - 2949 \beta_{8} + 11189 \beta_{6} + 447 \beta_{5} + \cdots + 854552 ) / 56 Copy content Toggle raw display
ν4\nu^{4}== (28507β11+13579β10+68330β8+60421β7+200669β6+66605268)/56 ( - 28507 \beta_{11} + 13579 \beta_{10} + 68330 \beta_{8} + 60421 \beta_{7} + 200669 \beta_{6} + \cdots - 66605268 ) / 56 Copy content Toggle raw display
ν5\nu^{5}== (214072β111639401β10+3846645β9+4442791β83846645β7++6463068)/56 ( 214072 \beta_{11} - 1639401 \beta_{10} + 3846645 \beta_{9} + 4442791 \beta_{8} - 3846645 \beta_{7} + \cdots + 6463068 ) / 56 Copy content Toggle raw display
ν6\nu^{6}== (52596103β11+24471648β10169236913β9376278467β8++152076711848)/56 ( 52596103 \beta_{11} + 24471648 \beta_{10} - 169236913 \beta_{9} - 376278467 \beta_{8} + \cdots + 152076711848 ) / 56 Copy content Toggle raw display
ν7\nu^{7}== (3985553385β11+3109587113β10+7837190810β8+9153261831β7+7267209008612)/56 ( - 3985553385 \beta_{11} + 3109587113 \beta_{10} + 7837190810 \beta_{8} + 9153261831 \beta_{7} + \cdots - 7267209008612 ) / 56 Copy content Toggle raw display
ν8\nu^{8}== (52708634544β11196514795911β10+439765022173β9+543735963837β8++905276874028)/56 ( 52708634544 \beta_{11} - 196514795911 \beta_{10} + 439765022173 \beta_{9} + 543735963837 \beta_{8} + \cdots + 905276874028 ) / 56 Copy content Toggle raw display
ν9\nu^{9}== (7515194137457β11+2420387138752β1022572086964135β947227483738197β8++18 ⁣ ⁣36)/56 ( 7515194137457 \beta_{11} + 2420387138752 \beta_{10} - 22572086964135 \beta_{9} - 47227483738197 \beta_{8} + \cdots + 18\!\cdots\!36 ) / 56 Copy content Toggle raw display
ν10\nu^{10}== (495573699183511β11+368700747752023β10+93 ⁣ ⁣92)/56 ( - 495573699183511 \beta_{11} + 368700747752023 \beta_{10} + \cdots - 93\!\cdots\!92 ) / 56 Copy content Toggle raw display
ν11\nu^{11}== (62 ⁣ ⁣68β11++11 ⁣ ⁣44)/56 ( 62\!\cdots\!68 \beta_{11} + \cdots + 11\!\cdots\!44 ) / 56 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/504Z)×\left(\mathbb{Z}/504\mathbb{Z}\right)^\times.

nn 7373 127127 253253 281281
χ(n)\chi(n) β1-\beta_{1} 11 11 11

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}
289.1
−2.74409 4.75291i
−25.0588 43.4031i
13.5741 + 23.5110i
−4.17642 7.23378i
10.9316 + 18.9342i
7.97360 + 13.8107i
−2.74409 + 4.75291i
−25.0588 + 43.4031i
13.5741 23.5110i
−4.17642 + 7.23378i
10.9316 18.9342i
7.97360 13.8107i
0 0 0 −45.7207 + 79.1905i 0 103.245 78.4062i 0 0 0
289.2 0 0 0 −23.1486 + 40.0946i 0 −126.558 + 28.1086i 0 0 0
289.3 0 0 0 −16.3419 + 28.3050i 0 23.3641 + 127.519i 0 0 0
289.4 0 0 0 0.634624 1.09920i 0 −62.9648 113.324i 0 0 0
289.5 0 0 0 30.2849 52.4550i 0 76.1691 + 104.906i 0 0 0
289.6 0 0 0 45.7916 79.3134i 0 58.7448 115.568i 0 0 0
361.1 0 0 0 −45.7207 79.1905i 0 103.245 + 78.4062i 0 0 0
361.2 0 0 0 −23.1486 40.0946i 0 −126.558 28.1086i 0 0 0
361.3 0 0 0 −16.3419 28.3050i 0 23.3641 127.519i 0 0 0
361.4 0 0 0 0.634624 + 1.09920i 0 −62.9648 + 113.324i 0 0 0
361.5 0 0 0 30.2849 + 52.4550i 0 76.1691 104.906i 0 0 0
361.6 0 0 0 45.7916 + 79.3134i 0 58.7448 + 115.568i 0 0 0
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.6.s.f 12
3.b odd 2 1 168.6.q.d 12
7.c even 3 1 inner 504.6.s.f 12
12.b even 2 1 336.6.q.n 12
21.h odd 6 1 168.6.q.d 12
84.n even 6 1 336.6.q.n 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
168.6.q.d 12 3.b odd 2 1
168.6.q.d 12 21.h odd 6 1
336.6.q.n 12 12.b even 2 1
336.6.q.n 12 84.n even 6 1
504.6.s.f 12 1.a even 1 1 trivial
504.6.s.f 12 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T512+17T511+11960T510+262771T59+112417320T58++94 ⁣ ⁣96 T_{5}^{12} + 17 T_{5}^{11} + 11960 T_{5}^{10} + 262771 T_{5}^{9} + 112417320 T_{5}^{8} + \cdots + 94\!\cdots\!96 acting on S6new(504,[χ])S_{6}^{\mathrm{new}}(504, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T12 T^{12} Copy content Toggle raw display
33 T12 T^{12} Copy content Toggle raw display
55 T12++94 ⁣ ⁣96 T^{12} + \cdots + 94\!\cdots\!96 Copy content Toggle raw display
77 T12++22 ⁣ ⁣49 T^{12} + \cdots + 22\!\cdots\!49 Copy content Toggle raw display
1111 T12++43 ⁣ ⁣56 T^{12} + \cdots + 43\!\cdots\!56 Copy content Toggle raw display
1313 (T6+59595303023616)2 (T^{6} + \cdots - 59595303023616)^{2} Copy content Toggle raw display
1717 T12++25 ⁣ ⁣00 T^{12} + \cdots + 25\!\cdots\!00 Copy content Toggle raw display
1919 T12++10 ⁣ ⁣16 T^{12} + \cdots + 10\!\cdots\!16 Copy content Toggle raw display
2323 T12++11 ⁣ ⁣00 T^{12} + \cdots + 11\!\cdots\!00 Copy content Toggle raw display
2929 (T6+28 ⁣ ⁣44)2 (T^{6} + \cdots - 28\!\cdots\!44)^{2} Copy content Toggle raw display
3131 T12++47 ⁣ ⁣25 T^{12} + \cdots + 47\!\cdots\!25 Copy content Toggle raw display
3737 T12++16 ⁣ ⁣64 T^{12} + \cdots + 16\!\cdots\!64 Copy content Toggle raw display
4141 (T6++17 ⁣ ⁣52)2 (T^{6} + \cdots + 17\!\cdots\!52)^{2} Copy content Toggle raw display
4343 (T6++10 ⁣ ⁣16)2 (T^{6} + \cdots + 10\!\cdots\!16)^{2} Copy content Toggle raw display
4747 T12++16 ⁣ ⁣00 T^{12} + \cdots + 16\!\cdots\!00 Copy content Toggle raw display
5353 T12++16 ⁣ ⁣64 T^{12} + \cdots + 16\!\cdots\!64 Copy content Toggle raw display
5959 T12++20 ⁣ ⁣00 T^{12} + \cdots + 20\!\cdots\!00 Copy content Toggle raw display
6161 T12++44 ⁣ ⁣00 T^{12} + \cdots + 44\!\cdots\!00 Copy content Toggle raw display
6767 T12++39 ⁣ ⁣96 T^{12} + \cdots + 39\!\cdots\!96 Copy content Toggle raw display
7171 (T6++24 ⁣ ⁣80)2 (T^{6} + \cdots + 24\!\cdots\!80)^{2} Copy content Toggle raw display
7373 T12++14 ⁣ ⁣56 T^{12} + \cdots + 14\!\cdots\!56 Copy content Toggle raw display
7979 T12++13 ⁣ ⁣81 T^{12} + \cdots + 13\!\cdots\!81 Copy content Toggle raw display
8383 (T6+70 ⁣ ⁣36)2 (T^{6} + \cdots - 70\!\cdots\!36)^{2} Copy content Toggle raw display
8989 T12++13 ⁣ ⁣16 T^{12} + \cdots + 13\!\cdots\!16 Copy content Toggle raw display
9797 (T6++65 ⁣ ⁣68)2 (T^{6} + \cdots + 65\!\cdots\!68)^{2} Copy content Toggle raw display
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