Properties

Label 845.2.c.g
Level $845$
Weight $2$
Character orbit 845.c
Analytic conductor $6.747$
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] = [845,2,Mod(506,845)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(845, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("845.506");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 845 = 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 845.c (of order \(2\), degree \(1\), not 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: \(6.74735897080\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.22581504.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} - 1) q^{3} + (\beta_{5} + \beta_{2} - 1) q^{4} + \beta_{6} q^{5} + ( - \beta_{7} + \beta_{6} + \cdots - \beta_1) q^{6} + ( - \beta_{7} + 2 \beta_{6} - \beta_{4}) q^{7}+ \cdots + (4 \beta_{6} - 2 \beta_{4} - 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} - 4 q^{4} + 8 q^{9} + 4 q^{10} + 20 q^{12} + 4 q^{14} + 4 q^{16} + 4 q^{17} + 24 q^{22} + 20 q^{23} - 8 q^{25} - 4 q^{27} + 16 q^{29} - 8 q^{30} - 20 q^{35} - 40 q^{36} - 16 q^{38} - 12 q^{40}+ \cdots + 32 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 5x^{6} + 2x^{5} - 11x^{4} + 4x^{3} + 20x^{2} - 32x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{7} + 4\nu^{6} - 5\nu^{5} - 2\nu^{4} + 11\nu^{3} - 4\nu^{2} - 28\nu + 32 ) / 8 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 2\nu^{6} - \nu^{5} + 4\nu^{4} - \nu^{3} - 6\nu^{2} + 10\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} + 8\nu^{6} - 3\nu^{5} - 10\nu^{4} + 13\nu^{3} + 8\nu^{2} - 32\nu + 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{7} + 12\nu^{6} - 5\nu^{5} - 18\nu^{4} + 19\nu^{3} + 28\nu^{2} - 64\nu + 40 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} + 9\nu^{6} - 5\nu^{5} - 13\nu^{4} + 21\nu^{3} + 13\nu^{2} - 54\nu + 40 ) / 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -4\nu^{7} + 11\nu^{6} - 6\nu^{5} - 17\nu^{4} + 24\nu^{3} + 15\nu^{2} - 62\nu + 48 ) / 4 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{7} + 7\nu^{6} - 3\nu^{5} - 11\nu^{4} + 15\nu^{3} + 11\nu^{2} - 40\nu + 30 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} + \beta_{5} + \beta_{4} + \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} - 2\beta_{6} + 2\beta_{5} - \beta_{4} + \beta_{3} + 2\beta_{2} - 2\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{7} - 5\beta_{6} + 3\beta_{5} + \beta_{4} + 2\beta_{3} + \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{7} - 4\beta_{6} + 4\beta_{5} - 3\beta_{4} + 2\beta_{3} - 3\beta_{2} - \beta _1 + 6 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -3\beta_{7} - 2\beta_{6} + 4\beta_{5} + 8\beta_{3} - 2\beta_{2} + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -8\beta_{7} + 4\beta_{6} + 8\beta_{5} - 2\beta_{4} + \beta_{3} - 5\beta_{2} - 3\beta _1 + 11 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/845\mathbb{Z}\right)^\times\).

\(n\) \(171\) \(677\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
506.1
0.665665 + 1.24775i
1.20036 + 0.747754i
−1.27597 + 0.609843i
1.40994 + 0.109843i
1.40994 0.109843i
−1.27597 0.609843i
1.20036 0.747754i
0.665665 1.24775i
2.49551i −2.82684 −4.22756 1.00000i 7.05440i 1.90521i 5.55889i 4.99102 2.49551
506.2 1.49551i 0.0947876 −0.236543 1.00000i 0.141756i 4.82684i 2.63726i −2.99102 −1.49551
506.3 1.21969i 2.33225 0.512364 1.00000i 2.84461i 3.60020i 3.06430i 2.43937 1.21969
506.4 0.219687i −1.60020 1.95174 1.00000i 0.351542i 0.332247i 0.868145i −0.439374 −0.219687
506.5 0.219687i −1.60020 1.95174 1.00000i 0.351542i 0.332247i 0.868145i −0.439374 −0.219687
506.6 1.21969i 2.33225 0.512364 1.00000i 2.84461i 3.60020i 3.06430i 2.43937 1.21969
506.7 1.49551i 0.0947876 −0.236543 1.00000i 0.141756i 4.82684i 2.63726i −2.99102 −1.49551
506.8 2.49551i −2.82684 −4.22756 1.00000i 7.05440i 1.90521i 5.55889i 4.99102 2.49551
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 506.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 845.2.c.g 8
13.b even 2 1 inner 845.2.c.g 8
13.c even 3 1 65.2.m.a 8
13.c even 3 1 845.2.m.g 8
13.d odd 4 1 845.2.a.l 4
13.d odd 4 1 845.2.a.m 4
13.e even 6 1 65.2.m.a 8
13.e even 6 1 845.2.m.g 8
13.f odd 12 2 845.2.e.m 8
13.f odd 12 2 845.2.e.n 8
39.f even 4 1 7605.2.a.cf 4
39.f even 4 1 7605.2.a.cj 4
39.h odd 6 1 585.2.bu.c 8
39.i odd 6 1 585.2.bu.c 8
52.i odd 6 1 1040.2.da.b 8
52.j odd 6 1 1040.2.da.b 8
65.g odd 4 1 4225.2.a.bi 4
65.g odd 4 1 4225.2.a.bl 4
65.l even 6 1 325.2.n.d 8
65.n even 6 1 325.2.n.d 8
65.q odd 12 1 325.2.m.b 8
65.q odd 12 1 325.2.m.c 8
65.r odd 12 1 325.2.m.b 8
65.r odd 12 1 325.2.m.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.m.a 8 13.c even 3 1
65.2.m.a 8 13.e even 6 1
325.2.m.b 8 65.q odd 12 1
325.2.m.b 8 65.r odd 12 1
325.2.m.c 8 65.q odd 12 1
325.2.m.c 8 65.r odd 12 1
325.2.n.d 8 65.l even 6 1
325.2.n.d 8 65.n even 6 1
585.2.bu.c 8 39.h odd 6 1
585.2.bu.c 8 39.i odd 6 1
845.2.a.l 4 13.d odd 4 1
845.2.a.m 4 13.d odd 4 1
845.2.c.g 8 1.a even 1 1 trivial
845.2.c.g 8 13.b even 2 1 inner
845.2.e.m 8 13.f odd 12 2
845.2.e.n 8 13.f odd 12 2
845.2.m.g 8 13.c even 3 1
845.2.m.g 8 13.e even 6 1
1040.2.da.b 8 52.i odd 6 1
1040.2.da.b 8 52.j odd 6 1
4225.2.a.bi 4 65.g odd 4 1
4225.2.a.bl 4 65.g odd 4 1
7605.2.a.cf 4 39.f even 4 1
7605.2.a.cj 4 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 10T_{2}^{6} + 27T_{2}^{4} + 22T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 10 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{4} + 2 T^{3} - 6 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$7$ \( T^{8} + 40 T^{6} + \cdots + 121 \) Copy content Toggle raw display
$11$ \( (T^{4} + 30 T^{2} + 33)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T^{4} - 2 T^{3} - 18 T^{2} + \cdots + 13)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 38 T^{2} + 169)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 10 T^{3} + \cdots - 299)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 8 T^{3} - 18 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 32 T^{2} + 64)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 112 T^{6} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( (T^{4} + 14 T^{2} + 1)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 2 T^{3} - 18 T^{2} + \cdots + 13)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 208 T^{6} + \cdots + 1763584 \) Copy content Toggle raw display
$53$ \( (T^{4} + 12 T^{3} + \cdots - 48)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + 84 T^{6} + \cdots + 9 \) Copy content Toggle raw display
$61$ \( (T^{4} - 28 T^{3} + \cdots + 1261)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 240 T^{6} + \cdots + 7667361 \) Copy content Toggle raw display
$71$ \( T^{8} + 436 T^{6} + \cdots + 109767529 \) Copy content Toggle raw display
$73$ \( T^{8} + 232 T^{6} + \cdots + 2930944 \) Copy content Toggle raw display
$79$ \( (T^{4} + 8 T^{3} + \cdots + 4432)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 192 T^{6} + \cdots + 36864 \) Copy content Toggle raw display
$89$ \( T^{8} + 612 T^{6} + \cdots + 78375609 \) Copy content Toggle raw display
$97$ \( T^{8} + 184 T^{6} + \cdots + 196249 \) Copy content Toggle raw display
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