Properties

Label 585.2.j.i
Level 585585
Weight 22
Character orbit 585.j
Analytic conductor 4.6714.671
Analytic rank 00
Dimension 1010
Inner twists 22

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [585,2,Mod(406,585)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(585, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("585.406");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: N N == 585=32513 585 = 3^{2} \cdot 5 \cdot 13
Weight: k k == 2 2
Character orbit: [χ][\chi] == 585.j (of order 33, degree 22, minimal)

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: no
Analytic conductor: 4.671248518244.67124851824
Analytic rank: 00
Dimension: 1010
Relative dimension: 55 over Q(ζ3)\Q(\zeta_{3})
Coefficient field: Q[x]/(x10)\mathbb{Q}[x]/(x^{10} - \cdots)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: x102x9+10x88x7+50x642x5+124x412x3+96x236x+36 x^{10} - 2x^{9} + 10x^{8} - 8x^{7} + 50x^{6} - 42x^{5} + 124x^{4} - 12x^{3} + 96x^{2} - 36x + 36 Copy content Toggle raw display
Coefficient ring: Z[a1,,a7]\Z[a_1, \ldots, a_{7}]
Coefficient ring index: 3 3
Twist minimal: yes
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

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,,β91,\beta_1,\ldots,\beta_{9} for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(β3β1)q2+(β8+β41)q4q5+β7q7+(β6+β5+β3+1)q8+(β3+β1)q10+(β9+β7+β4+β1)q11++(3β8+2β7++7)q98+O(q100) q + ( - \beta_{3} - \beta_1) q^{2} + ( - \beta_{8} + \beta_{4} - 1) q^{4} - q^{5} + \beta_{7} q^{7} + (\beta_{6} + \beta_{5} + \beta_{3} + \cdots - 1) q^{8} + (\beta_{3} + \beta_1) q^{10} + (\beta_{9} + \beta_{7} + \beta_{4} + \cdots - \beta_1) q^{11}+ \cdots + ( - 3 \beta_{8} + 2 \beta_{7} + \cdots + 7) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 10q+2q26q410q5q712q82q10+8q11+q13+8q144q16+4q19+6q2014q226q23+10q25+10q26+2q28+16q29++30q98+O(q100) 10 q + 2 q^{2} - 6 q^{4} - 10 q^{5} - q^{7} - 12 q^{8} - 2 q^{10} + 8 q^{11} + q^{13} + 8 q^{14} - 4 q^{16} + 4 q^{19} + 6 q^{20} - 14 q^{22} - 6 q^{23} + 10 q^{25} + 10 q^{26} + 2 q^{28} + 16 q^{29}+ \cdots + 30 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x102x9+10x88x7+50x642x5+124x412x3+96x236x+36 x^{10} - 2x^{9} + 10x^{8} - 8x^{7} + 50x^{6} - 42x^{5} + 124x^{4} - 12x^{3} + 96x^{2} - 36x + 36 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (355ν92199ν8+7398ν718528ν6+39184ν587134ν4+55404)/40746 ( 355 \nu^{9} - 2199 \nu^{8} + 7398 \nu^{7} - 18528 \nu^{6} + 39184 \nu^{5} - 87134 \nu^{4} + \cdots - 55404 ) / 40746 Copy content Toggle raw display
β3\beta_{3}== (4282ν912311ν8+50191ν764307ν6+231068ν5318192ν4+235800)/448206 ( 4282 \nu^{9} - 12311 \nu^{8} + 50191 \nu^{7} - 64307 \nu^{6} + 231068 \nu^{5} - 318192 \nu^{4} + \cdots - 235800 ) / 448206 Copy content Toggle raw display
β4\beta_{4}== (6550ν9+8818ν853189ν7+2209ν6263193ν5+44032ν4++99432)/448206 ( - 6550 \nu^{9} + 8818 \nu^{8} - 53189 \nu^{7} + 2209 \nu^{6} - 263193 \nu^{5} + 44032 \nu^{4} + \cdots + 99432 ) / 448206 Copy content Toggle raw display
β5\beta_{5}== (8029ν9+19682ν880242ν7+81275ν6369416ν5+508704ν4++1426266)/448206 ( - 8029 \nu^{9} + 19682 \nu^{8} - 80242 \nu^{7} + 81275 \nu^{6} - 369416 \nu^{5} + 508704 \nu^{4} + \cdots + 1426266 ) / 448206 Copy content Toggle raw display
β6\beta_{6}== (9476ν9+17684ν889335ν7+36452ν6354900ν5+123782ν4+408504)/448206 ( - 9476 \nu^{9} + 17684 \nu^{8} - 89335 \nu^{7} + 36452 \nu^{6} - 354900 \nu^{5} + 123782 \nu^{4} + \cdots - 408504 ) / 448206 Copy content Toggle raw display
β7\beta_{7}== (10145ν97555ν872631ν7139765ν6519015ν5700936ν4+953850)/448206 ( - 10145 \nu^{9} - 7555 \nu^{8} - 72631 \nu^{7} - 139765 \nu^{6} - 519015 \nu^{5} - 700936 \nu^{4} + \cdots - 953850 ) / 448206 Copy content Toggle raw display
β8\beta_{8}== (5301ν9+6361ν843172ν73447ν6217077ν519472ν4+297390)/149402 ( - 5301 \nu^{9} + 6361 \nu^{8} - 43172 \nu^{7} - 3447 \nu^{6} - 217077 \nu^{5} - 19472 \nu^{4} + \cdots - 297390 ) / 149402 Copy content Toggle raw display
β9\beta_{9}== (9385ν9+23127ν894287ν7+111339ν6434076ν5+597744ν4++675982)/149402 ( - 9385 \nu^{9} + 23127 \nu^{8} - 94287 \nu^{7} + 111339 \nu^{6} - 434076 \nu^{5} + 597744 \nu^{4} + \cdots + 675982 ) / 149402 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== β8+β53β4+β3 \beta_{8} + \beta_{5} - 3\beta_{4} + \beta_{3} Copy content Toggle raw display
ν3\nu^{3}== β6+β5+5β3β21 \beta_{6} + \beta_{5} + 5\beta_{3} - \beta_{2} - 1 Copy content Toggle raw display
ν4\nu^{4}== 7β8+β7+β6+14β4β114 -7\beta_{8} + \beta_{7} + \beta_{6} + 14\beta_{4} - \beta _1 - 14 Copy content Toggle raw display
ν5\nu^{5}== 2β910β8+2β710β5+10β430β3+8β220β1 2\beta_{9} - 10\beta_{8} + 2\beta_{7} - 10\beta_{5} + 10\beta_{4} - 30\beta_{3} + 8\beta_{2} - 20\beta_1 Copy content Toggle raw display
ν6\nu^{6}== 10β912β646β558β3+12β2+76 10\beta_{9} - 12\beta_{6} - 46\beta_{5} - 58\beta_{3} + 12\beta_{2} + 76 Copy content Toggle raw display
ν7\nu^{7}== 82β822β756β684β4+110β1+84 82\beta_{8} - 22\beta_{7} - 56\beta_{6} - 84\beta_{4} + 110\beta _1 + 84 Copy content Toggle raw display
ν8\nu^{8}== 78β9+304β878β7+304β5446β4+414β3104β2+110β1 -78\beta_{9} + 304\beta_{8} - 78\beta_{7} + 304\beta_{5} - 446\beta_{4} + 414\beta_{3} - 104\beta_{2} + 110\beta_1 Copy content Toggle raw display
ν9\nu^{9}== 182β9+382β6+622β5+1268β3382β2660 -182\beta_{9} + 382\beta_{6} + 622\beta_{5} + 1268\beta_{3} - 382\beta_{2} - 660 Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/585Z)×\left(\mathbb{Z}/585\mathbb{Z}\right)^\times.

nn 326326 352352 496496
χ(n)\chi(n) 11 11 1+β4-1 + \beta_{4}

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}
406.1
−1.06141 + 1.83842i
−0.473183 + 0.819577i
0.313396 0.542817i
0.905157 1.56778i
1.31604 2.27945i
−1.06141 1.83842i
−0.473183 0.819577i
0.313396 + 0.542817i
0.905157 + 1.56778i
1.31604 + 2.27945i
−1.06141 1.83842i 0 −1.25320 + 2.17061i −1.00000 0 −0.733534 + 1.27052i 1.07500 0 1.06141 + 1.83842i
406.2 −0.473183 0.819577i 0 0.552196 0.956432i −1.00000 0 0.781529 1.35365i −2.93789 0 0.473183 + 0.819577i
406.3 0.313396 + 0.542817i 0 0.803566 1.39182i −1.00000 0 −2.21563 + 3.83759i 2.26092 0 −0.313396 0.542817i
406.4 0.905157 + 1.56778i 0 −0.638619 + 1.10612i −1.00000 0 2.21251 3.83219i 1.30843 0 −0.905157 1.56778i
406.5 1.31604 + 2.27945i 0 −2.46394 + 4.26767i −1.00000 0 −0.544875 + 0.943751i −7.70645 0 −1.31604 2.27945i
451.1 −1.06141 + 1.83842i 0 −1.25320 2.17061i −1.00000 0 −0.733534 1.27052i 1.07500 0 1.06141 1.83842i
451.2 −0.473183 + 0.819577i 0 0.552196 + 0.956432i −1.00000 0 0.781529 + 1.35365i −2.93789 0 0.473183 0.819577i
451.3 0.313396 0.542817i 0 0.803566 + 1.39182i −1.00000 0 −2.21563 3.83759i 2.26092 0 −0.313396 + 0.542817i
451.4 0.905157 1.56778i 0 −0.638619 1.10612i −1.00000 0 2.21251 + 3.83219i 1.30843 0 −0.905157 + 1.56778i
451.5 1.31604 2.27945i 0 −2.46394 4.26767i −1.00000 0 −0.544875 0.943751i −7.70645 0 −1.31604 + 2.27945i
nn: e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 406.5
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 585.2.j.i yes 10
3.b odd 2 1 585.2.j.h 10
13.c even 3 1 inner 585.2.j.i yes 10
13.c even 3 1 7605.2.a.cm 5
13.e even 6 1 7605.2.a.cn 5
39.h odd 6 1 7605.2.a.cl 5
39.i odd 6 1 585.2.j.h 10
39.i odd 6 1 7605.2.a.co 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.j.h 10 3.b odd 2 1
585.2.j.h 10 39.i odd 6 1
585.2.j.i yes 10 1.a even 1 1 trivial
585.2.j.i yes 10 13.c even 3 1 inner
7605.2.a.cl 5 39.h odd 6 1
7605.2.a.cm 5 13.c even 3 1
7605.2.a.cn 5 13.e even 6 1
7605.2.a.co 5 39.i odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T2102T29+10T288T27+50T2642T25+124T2412T23+96T2236T2+36 T_{2}^{10} - 2T_{2}^{9} + 10T_{2}^{8} - 8T_{2}^{7} + 50T_{2}^{6} - 42T_{2}^{5} + 124T_{2}^{4} - 12T_{2}^{3} + 96T_{2}^{2} - 36T_{2} + 36 acting on S2new(585,[χ])S_{2}^{\mathrm{new}}(585, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T102T9++36 T^{10} - 2 T^{9} + \cdots + 36 Copy content Toggle raw display
33 T10 T^{10} Copy content Toggle raw display
55 (T+1)10 (T + 1)^{10} Copy content Toggle raw display
77 T10+T9++2401 T^{10} + T^{9} + \cdots + 2401 Copy content Toggle raw display
1111 T108T9++576 T^{10} - 8 T^{9} + \cdots + 576 Copy content Toggle raw display
1313 T10T9++371293 T^{10} - T^{9} + \cdots + 371293 Copy content Toggle raw display
1717 T10+50T8++412164 T^{10} + 50 T^{8} + \cdots + 412164 Copy content Toggle raw display
1919 T104T9++144 T^{10} - 4 T^{9} + \cdots + 144 Copy content Toggle raw display
2323 T10+6T9++5184 T^{10} + 6 T^{9} + \cdots + 5184 Copy content Toggle raw display
2929 T1016T9++66564 T^{10} - 16 T^{9} + \cdots + 66564 Copy content Toggle raw display
3131 (T59T4++603)2 (T^{5} - 9 T^{4} + \cdots + 603)^{2} Copy content Toggle raw display
3737 T104T9++20736 T^{10} - 4 T^{9} + \cdots + 20736 Copy content Toggle raw display
4141 T10++141562404 T^{10} + \cdots + 141562404 Copy content Toggle raw display
4343 T10+15T9++4239481 T^{10} + 15 T^{9} + \cdots + 4239481 Copy content Toggle raw display
4747 (T5+10T4++7434)2 (T^{5} + 10 T^{4} + \cdots + 7434)^{2} Copy content Toggle raw display
5353 (T520T4++15552)2 (T^{5} - 20 T^{4} + \cdots + 15552)^{2} Copy content Toggle raw display
5959 T1012T9++7584516 T^{10} - 12 T^{9} + \cdots + 7584516 Copy content Toggle raw display
6161 T10+11T9++9138529 T^{10} + 11 T^{9} + \cdots + 9138529 Copy content Toggle raw display
6767 T10+5T9++289 T^{10} + 5 T^{9} + \cdots + 289 Copy content Toggle raw display
7171 T1010T9++26244 T^{10} - 10 T^{9} + \cdots + 26244 Copy content Toggle raw display
7373 (T5T4236T3++9693)2 (T^{5} - T^{4} - 236 T^{3} + \cdots + 9693)^{2} Copy content Toggle raw display
7979 (T5+17T4++2349)2 (T^{5} + 17 T^{4} + \cdots + 2349)^{2} Copy content Toggle raw display
8383 (T5+16T4++6264)2 (T^{5} + 16 T^{4} + \cdots + 6264)^{2} Copy content Toggle raw display
8989 T104T9++2916 T^{10} - 4 T^{9} + \cdots + 2916 Copy content Toggle raw display
9797 T1011T9++58967041 T^{10} - 11 T^{9} + \cdots + 58967041 Copy content Toggle raw display
show more
show less