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Properties

Label 585.2.j.h
Level $585$
Weight $2$
Character orbit 585.j
Analytic conductor $4.671$
Analytic rank $0$
Dimension $10$
Inner twists $2$

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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: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.j (of order \(3\), degree \(2\), 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: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2x^{9} + 10x^{8} - 8x^{7} + 50x^{6} - 42x^{5} + 124x^{4} - 12x^{3} + 96x^{2} - 36x + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{3} + \beta_1) q^{2} + ( - \beta_{8} + \beta_{4} - 1) q^{4} + q^{5} + \beta_{7} q^{7} + ( - \beta_{6} - \beta_{5} - \beta_{3} + \cdots + 1) q^{8} + (\beta_{3} + \beta_1) q^{10} + ( - \beta_{9} - \beta_{7} + \cdots + \beta_1) q^{11}+ \cdots + (3 \beta_{8} - 2 \beta_{7} - 3 \beta_{6} + \cdots - 7) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 2 q^{2} - 6 q^{4} + 10 q^{5} - q^{7} + 12 q^{8} - 2 q^{10} - 8 q^{11} + q^{13} - 8 q^{14} - 4 q^{16} + 4 q^{19} - 6 q^{20} - 14 q^{22} + 6 q^{23} + 10 q^{25} - 10 q^{26} + 2 q^{28} - 16 q^{29}+ \cdots - 30 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 355 \nu^{9} - 2199 \nu^{8} + 7398 \nu^{7} - 18528 \nu^{6} + 39184 \nu^{5} - 87134 \nu^{4} + \cdots - 55404 ) / 40746 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4282 \nu^{9} - 12311 \nu^{8} + 50191 \nu^{7} - 64307 \nu^{6} + 231068 \nu^{5} - 318192 \nu^{4} + \cdots - 235800 ) / 448206 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 6550 \nu^{9} + 8818 \nu^{8} - 53189 \nu^{7} + 2209 \nu^{6} - 263193 \nu^{5} + 44032 \nu^{4} + \cdots + 99432 ) / 448206 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 8029 \nu^{9} + 19682 \nu^{8} - 80242 \nu^{7} + 81275 \nu^{6} - 369416 \nu^{5} + 508704 \nu^{4} + \cdots + 1426266 ) / 448206 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 9476 \nu^{9} + 17684 \nu^{8} - 89335 \nu^{7} + 36452 \nu^{6} - 354900 \nu^{5} + 123782 \nu^{4} + \cdots - 408504 ) / 448206 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10145 \nu^{9} - 7555 \nu^{8} - 72631 \nu^{7} - 139765 \nu^{6} - 519015 \nu^{5} - 700936 \nu^{4} + \cdots - 953850 ) / 448206 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 5301 \nu^{9} + 6361 \nu^{8} - 43172 \nu^{7} - 3447 \nu^{6} - 217077 \nu^{5} - 19472 \nu^{4} + \cdots - 297390 ) / 149402 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 9385 \nu^{9} + 23127 \nu^{8} - 94287 \nu^{7} + 111339 \nu^{6} - 434076 \nu^{5} + 597744 \nu^{4} + \cdots + 675982 ) / 149402 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} + \beta_{5} - 3\beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + 5\beta_{3} - \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{8} + \beta_{7} + \beta_{6} + 14\beta_{4} - \beta _1 - 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{9} - 10\beta_{8} + 2\beta_{7} - 10\beta_{5} + 10\beta_{4} - 30\beta_{3} + 8\beta_{2} - 20\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10\beta_{9} - 12\beta_{6} - 46\beta_{5} - 58\beta_{3} + 12\beta_{2} + 76 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 82\beta_{8} - 22\beta_{7} - 56\beta_{6} - 84\beta_{4} + 110\beta _1 + 84 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -78\beta_{9} + 304\beta_{8} - 78\beta_{7} + 304\beta_{5} - 446\beta_{4} + 414\beta_{3} - 104\beta_{2} + 110\beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( -182\beta_{9} + 382\beta_{6} + 622\beta_{5} + 1268\beta_{3} - 382\beta_{2} - 660 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(326\) \(352\) \(496\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{4}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
406.1
1.31604 − 2.27945i
0.905157 − 1.56778i
0.313396 − 0.542817i
−0.473183 + 0.819577i
−1.06141 + 1.83842i
1.31604 + 2.27945i
0.905157 + 1.56778i
0.313396 + 0.542817i
−0.473183 − 0.819577i
−1.06141 − 1.83842i
−1.31604 − 2.27945i 0 −2.46394 + 4.26767i 1.00000 0 −0.544875 + 0.943751i 7.70645 0 −1.31604 − 2.27945i
406.2 −0.905157 − 1.56778i 0 −0.638619 + 1.10612i 1.00000 0 2.21251 − 3.83219i −1.30843 0 −0.905157 − 1.56778i
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 0.473183 + 0.819577i 0 0.552196 − 0.956432i 1.00000 0 0.781529 − 1.35365i 2.93789 0 0.473183 + 0.819577i
406.5 1.06141 + 1.83842i 0 −1.25320 + 2.17061i 1.00000 0 −0.733534 + 1.27052i −1.07500 0 1.06141 + 1.83842i
451.1 −1.31604 + 2.27945i 0 −2.46394 − 4.26767i 1.00000 0 −0.544875 − 0.943751i 7.70645 0 −1.31604 + 2.27945i
451.2 −0.905157 + 1.56778i 0 −0.638619 − 1.10612i 1.00000 0 2.21251 + 3.83219i −1.30843 0 −0.905157 + 1.56778i
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 0.473183 − 0.819577i 0 0.552196 + 0.956432i 1.00000 0 0.781529 + 1.35365i 2.93789 0 0.473183 − 0.819577i
451.5 1.06141 − 1.83842i 0 −1.25320 − 2.17061i 1.00000 0 −0.733534 − 1.27052i −1.07500 0 1.06141 − 1.83842i
\(n\): 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.h ✓ 10
3.b odd 2 1 585.2.j.i yes 10
13.c even 3 1 inner 585.2.j.h ✓ 10
13.c even 3 1 7605.2.a.co 5
13.e even 6 1 7605.2.a.cl 5
39.h odd 6 1 7605.2.a.cn 5
39.i odd 6 1 585.2.j.i yes 10
39.i odd 6 1 7605.2.a.cm 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
585.2.j.h ✓ 10 1.a even 1 1 trivial
585.2.j.h ✓ 10 13.c even 3 1 inner
585.2.j.i yes 10 3.b odd 2 1
585.2.j.i yes 10 39.i odd 6 1
7605.2.a.cl 5 13.e even 6 1
7605.2.a.cm 5 39.i odd 6 1
7605.2.a.cn 5 39.h odd 6 1
7605.2.a.co 5 13.c even 3 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 2 T^{9} + \cdots + 36 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( (T - 1)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} + T^{9} + \cdots + 2401 \) Copy content Toggle raw display
$11$ \( T^{10} + 8 T^{9} + \cdots + 576 \) Copy content Toggle raw display
$13$ \( T^{10} - T^{9} + \cdots + 371293 \) Copy content Toggle raw display
$17$ \( T^{10} + 50 T^{8} + \cdots + 412164 \) Copy content Toggle raw display
$19$ \( T^{10} - 4 T^{9} + \cdots + 144 \) Copy content Toggle raw display
$23$ \( T^{10} - 6 T^{9} + \cdots + 5184 \) Copy content Toggle raw display
$29$ \( T^{10} + 16 T^{9} + \cdots + 66564 \) Copy content Toggle raw display
$31$ \( (T^{5} - 9 T^{4} + \cdots + 603)^{2} \) Copy content Toggle raw display
$37$ \( T^{10} - 4 T^{9} + \cdots + 20736 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 141562404 \) Copy content Toggle raw display
$43$ \( T^{10} + 15 T^{9} + \cdots + 4239481 \) Copy content Toggle raw display
$47$ \( (T^{5} - 10 T^{4} + \cdots - 7434)^{2} \) Copy content Toggle raw display
$53$ \( (T^{5} + 20 T^{4} + \cdots - 15552)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + 12 T^{9} + \cdots + 7584516 \) Copy content Toggle raw display
$61$ \( T^{10} + 11 T^{9} + \cdots + 9138529 \) Copy content Toggle raw display
$67$ \( T^{10} + 5 T^{9} + \cdots + 289 \) Copy content Toggle raw display
$71$ \( T^{10} + 10 T^{9} + \cdots + 26244 \) Copy content Toggle raw display
$73$ \( (T^{5} - T^{4} - 236 T^{3} + \cdots + 9693)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + 17 T^{4} + \cdots + 2349)^{2} \) Copy content Toggle raw display
$83$ \( (T^{5} - 16 T^{4} + \cdots - 6264)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + 4 T^{9} + \cdots + 2916 \) Copy content Toggle raw display
$97$ \( T^{10} - 11 T^{9} + \cdots + 58967041 \) Copy content Toggle raw display
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