Properties

Label 855.2.k.h
Level $855$
Weight $2$
Character orbit 855.k
Analytic conductor $6.827$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [855,2,Mod(406,855)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(855, 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("855.406");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 855 = 3^{2} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 855.k (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: \(6.82720937282\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.4601315889.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 6x^{6} - 3x^{5} + 26x^{4} - 14x^{3} + 31x^{2} + 12x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{7} q^{2} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \cdots - 1) q^{4}+ \cdots + ( - 2 \beta_{6} + \beta_{4} - \beta_{3} + \cdots - 3) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{2} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \cdots - 1) q^{4}+ \cdots + (\beta_{7} + 3 \beta_{5} + \cdots - 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} - 5 q^{4} + 4 q^{5} - 8 q^{7} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} - 5 q^{4} + 4 q^{5} - 8 q^{7} - 24 q^{8} - q^{10} + 4 q^{11} - 7 q^{13} - q^{14} - 7 q^{16} - q^{17} + 5 q^{19} - 10 q^{20} - 2 q^{22} + 2 q^{23} - 4 q^{25} - 6 q^{26} + 19 q^{28} - q^{29} + 30 q^{32} - 15 q^{34} - 4 q^{35} - 4 q^{37} - 13 q^{38} - 12 q^{40} - 8 q^{41} - q^{43} - 12 q^{44} + 24 q^{46} - 12 q^{47} - 20 q^{49} - 2 q^{50} + 3 q^{52} - 5 q^{53} + 2 q^{55} + 82 q^{56} - 54 q^{58} - 5 q^{59} + 37 q^{62} + 112 q^{64} - 14 q^{65} - 4 q^{67} - 32 q^{68} + q^{70} + 20 q^{71} + 20 q^{73} + 25 q^{74} + 63 q^{76} - 28 q^{77} - 17 q^{79} + 7 q^{80} - 21 q^{82} - 2 q^{83} + q^{85} + 8 q^{86} - 14 q^{88} + 11 q^{89} - 6 q^{91} - q^{92} - 62 q^{94} + 4 q^{95} - q^{97} + 9 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 6x^{6} - 3x^{5} + 26x^{4} - 14x^{3} + 31x^{2} + 12x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -26\nu^{7} - 189\nu^{6} + 729\nu^{5} - 911\nu^{4} + 3051\nu^{3} - 3618\nu^{2} + 14317\nu - 1215 ) / 4243 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 115\nu^{7} + 20\nu^{6} + 529\nu^{5} + 276\nu^{4} + 3314\nu^{3} + 989\nu^{2} + 483\nu + 1947 ) / 4243 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -135\nu^{7} + 161\nu^{6} - 621\nu^{5} - 324\nu^{4} - 2599\nu^{3} - 1161\nu^{2} - 567\nu - 11694 ) / 4243 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -649\nu^{7} + 994\nu^{6} - 3834\nu^{5} + 3534\nu^{4} - 16046\nu^{3} + 19028\nu^{2} - 17152\nu + 6390 ) / 12729 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 434\nu^{7} - 109\nu^{6} + 2845\nu^{5} + 193\nu^{4} + 12064\nu^{3} + 338\nu^{2} + 16249\nu + 6573 ) / 4243 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 514\nu^{7} - 833\nu^{6} + 3213\nu^{5} - 3858\nu^{4} + 13447\nu^{3} - 15946\nu^{2} + 16585\nu - 5355 ) / 4243 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + 3\beta_{5} - \beta_{4} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{6} + 4\beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -5\beta_{7} - 12\beta_{5} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{7} + 6\beta_{6} + \beta_{4} - 17\beta_{3} - 17\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7\beta_{6} + 23\beta_{4} - \beta_{3} - 7\beta_{2} + 51 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 8\beta_{7} + 3\beta_{5} - 30\beta_{2} + 74\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/855\mathbb{Z}\right)^\times\).

\(n\) \(172\) \(191\) \(496\)
\(\chi(n)\) \(1\) \(1\) \(-1 + \beta_{5}\)

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
1.07988 + 1.87040i
−1.02359 1.77290i
0.689667 + 1.19454i
−0.245959 0.426014i
1.07988 1.87040i
−1.02359 + 1.77290i
0.689667 1.19454i
−0.245959 + 0.426014i
−0.832272 1.44154i 0 −0.385355 + 0.667454i 0.500000 + 0.866025i 0 −2.43525 −2.04621 0 0.832272 1.44154i
406.2 −0.595455 1.03136i 0 0.290867 0.503797i 0.500000 + 0.866025i 0 −0.609175 −3.07461 0 0.595455 1.03136i
406.3 0.548719 + 0.950409i 0 0.397815 0.689035i 0.500000 + 0.866025i 0 1.89307 3.06803 0 −0.548719 + 0.950409i
406.4 1.37901 + 2.38851i 0 −2.80333 + 4.85550i 0.500000 + 0.866025i 0 −2.84864 −9.94721 0 −1.37901 + 2.38851i
676.1 −0.832272 + 1.44154i 0 −0.385355 0.667454i 0.500000 0.866025i 0 −2.43525 −2.04621 0 0.832272 + 1.44154i
676.2 −0.595455 + 1.03136i 0 0.290867 + 0.503797i 0.500000 0.866025i 0 −0.609175 −3.07461 0 0.595455 + 1.03136i
676.3 0.548719 0.950409i 0 0.397815 + 0.689035i 0.500000 0.866025i 0 1.89307 3.06803 0 −0.548719 0.950409i
676.4 1.37901 2.38851i 0 −2.80333 4.85550i 0.500000 0.866025i 0 −2.84864 −9.94721 0 −1.37901 2.38851i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 406.4
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 855.2.k.h 8
3.b odd 2 1 95.2.e.c 8
12.b even 2 1 1520.2.q.o 8
15.d odd 2 1 475.2.e.e 8
15.e even 4 2 475.2.j.c 16
19.c even 3 1 inner 855.2.k.h 8
57.f even 6 1 1805.2.a.i 4
57.h odd 6 1 95.2.e.c 8
57.h odd 6 1 1805.2.a.o 4
228.m even 6 1 1520.2.q.o 8
285.n odd 6 1 475.2.e.e 8
285.n odd 6 1 9025.2.a.bg 4
285.q even 6 1 9025.2.a.bp 4
285.v even 12 2 475.2.j.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.e.c 8 3.b odd 2 1
95.2.e.c 8 57.h odd 6 1
475.2.e.e 8 15.d odd 2 1
475.2.e.e 8 285.n odd 6 1
475.2.j.c 16 15.e even 4 2
475.2.j.c 16 285.v even 12 2
855.2.k.h 8 1.a even 1 1 trivial
855.2.k.h 8 19.c even 3 1 inner
1520.2.q.o 8 12.b even 2 1
1520.2.q.o 8 228.m even 6 1
1805.2.a.i 4 57.f even 6 1
1805.2.a.o 4 57.h odd 6 1
9025.2.a.bg 4 285.n odd 6 1
9025.2.a.bp 4 285.q even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(855, [\chi])\):

\( T_{2}^{8} - T_{2}^{7} + 7T_{2}^{6} + 4T_{2}^{5} + 31T_{2}^{4} + 6T_{2}^{3} + 37T_{2}^{2} + 6T_{2} + 36 \) Copy content Toggle raw display
\( T_{7}^{4} + 4T_{7}^{3} - T_{7}^{2} - 15T_{7} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - T^{7} + \cdots + 36 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{2} - T + 1)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 4 T^{3} - T^{2} + \cdots - 8)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} - 2 T^{3} - 25 T^{2} + \cdots + 3)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 7 T^{7} + \cdots + 256 \) Copy content Toggle raw display
$17$ \( T^{8} + T^{7} + \cdots + 11664 \) Copy content Toggle raw display
$19$ \( T^{8} - 5 T^{7} + \cdots + 130321 \) Copy content Toggle raw display
$23$ \( T^{8} - 2 T^{7} + \cdots + 36 \) Copy content Toggle raw display
$29$ \( T^{8} + T^{7} + \cdots + 19881 \) Copy content Toggle raw display
$31$ \( (T^{4} - 67 T^{2} + \cdots + 1063)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 2 T^{3} + \cdots - 118)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 8 T^{7} + \cdots + 5008644 \) Copy content Toggle raw display
$43$ \( T^{8} + T^{7} + \cdots + 630436 \) Copy content Toggle raw display
$47$ \( T^{8} + 12 T^{7} + \cdots + 5363856 \) Copy content Toggle raw display
$53$ \( T^{8} + 5 T^{7} + \cdots + 2916 \) Copy content Toggle raw display
$59$ \( T^{8} + 5 T^{7} + \cdots + 3515625 \) Copy content Toggle raw display
$61$ \( T^{8} + 130 T^{6} + \cdots + 9296401 \) Copy content Toggle raw display
$67$ \( T^{8} + 4 T^{7} + \cdots + 4096 \) Copy content Toggle raw display
$71$ \( T^{8} - 20 T^{7} + \cdots + 59049 \) Copy content Toggle raw display
$73$ \( T^{8} - 20 T^{7} + \cdots + 2979076 \) Copy content Toggle raw display
$79$ \( T^{8} + 17 T^{7} + \cdots + 33856 \) Copy content Toggle raw display
$83$ \( (T^{4} + T^{3} - 62 T^{2} + \cdots + 366)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} - 11 T^{7} + \cdots + 14561856 \) Copy content Toggle raw display
$97$ \( T^{8} + T^{7} + \cdots + 55383364 \) Copy content Toggle raw display
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