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Properties

Label 845.2.c.f
Level $845$
Weight $2$
Character orbit 845.c
Analytic conductor $6.747$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: 6.0.153664.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{4} - \beta_{2} - 1) q^{3} + \beta_{2} q^{4} - \beta_{5} q^{5} + ( - \beta_{5} - \beta_1) q^{6} + ( - 2 \beta_{5} - \beta_{3}) q^{7} + (\beta_{3} + \beta_1) q^{8} + (3 \beta_{4} + 4 \beta_{2} - 1) q^{9}+ \cdots + (7 \beta_{5} - 4 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 10 q^{3} + 2 q^{4} + 8 q^{9} + 2 q^{10} - 8 q^{12} + 8 q^{14} - 10 q^{16} + 6 q^{17} - 12 q^{22} + 4 q^{23} - 6 q^{25} - 16 q^{27} - 12 q^{29} - 8 q^{30} - 10 q^{35} + 26 q^{36} + 38 q^{38} - 18 q^{42}+ \cdots - 16 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 5x^{4} + 6x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + 3\nu^{2} + 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} + 4\nu^{3} + 3\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - 3\beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 4\beta_{3} + 9\beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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
− 1.80194i
− 1.24698i
− 0.445042i
0.445042i
1.24698i
1.80194i
− 1.80194i −1.55496 −1.24698 1.00000i 2.80194i 1.55496i − 1.35690i −0.582105 1.80194
506.2 − 1.24698i −0.198062 0.445042 − 1.00000i 0.246980i − 0.198062i − 3.04892i −2.96077 −1.24698
506.3 − 0.445042i −3.24698 1.80194 1.00000i 1.44504i 3.24698i − 1.69202i 7.54288 0.445042
506.4 0.445042i −3.24698 1.80194 − 1.00000i − 1.44504i − 3.24698i 1.69202i 7.54288 0.445042
506.5 1.24698i −0.198062 0.445042 1.00000i − 0.246980i 0.198062i 3.04892i −2.96077 −1.24698
506.6 1.80194i −1.55496 −1.24698 − 1.00000i − 2.80194i − 1.55496i 1.35690i −0.582105 1.80194
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 506.6
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.f 6
13.b even 2 1 inner 845.2.c.f 6
13.c even 3 2 845.2.m.i 12
13.d odd 4 1 845.2.a.h ✓ 3
13.d odd 4 1 845.2.a.j yes 3
13.e even 6 2 845.2.m.i 12
13.f odd 12 2 845.2.e.j 6
13.f odd 12 2 845.2.e.l 6
39.f even 4 1 7605.2.a.br 3
39.f even 4 1 7605.2.a.by 3
65.g odd 4 1 4225.2.a.bd 3
65.g odd 4 1 4225.2.a.bf 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
845.2.a.h ✓ 3 13.d odd 4 1
845.2.a.j yes 3 13.d odd 4 1
845.2.c.f 6 1.a even 1 1 trivial
845.2.c.f 6 13.b even 2 1 inner
845.2.e.j 6 13.f odd 12 2
845.2.e.l 6 13.f odd 12 2
845.2.m.i 12 13.c even 3 2
845.2.m.i 12 13.e even 6 2
4225.2.a.bd 3 65.g odd 4 1
4225.2.a.bf 3 65.g odd 4 1
7605.2.a.br 3 39.f even 4 1
7605.2.a.by 3 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 5T_{2}^{4} + 6T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 5 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{3} + 5 T^{2} + 6 T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{2} + 1)^{3} \) Copy content Toggle raw display
$7$ \( T^{6} + 13 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{6} + 17 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$13$ \( T^{6} \) Copy content Toggle raw display
$17$ \( (T^{3} - 3 T^{2} - 18 T + 27)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} + 82 T^{4} + \cdots + 841 \) Copy content Toggle raw display
$23$ \( (T^{3} - 2 T^{2} - 57 T + 71)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + 6 T^{2} + \cdots - 307)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 101 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$37$ \( T^{6} + 27 T^{4} + \cdots + 169 \) Copy content Toggle raw display
$41$ \( T^{6} + 194 T^{4} + \cdots + 1849 \) Copy content Toggle raw display
$43$ \( (T^{3} - 23 T^{2} + \cdots - 433)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} + 146 T^{4} + \cdots + 1681 \) Copy content Toggle raw display
$53$ \( (T^{3} + 2 T^{2} - T - 1)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} + 59 T^{4} + \cdots + 169 \) Copy content Toggle raw display
$61$ \( (T^{3} + 17 T^{2} + \cdots + 169)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 243 T^{4} + \cdots + 123201 \) Copy content Toggle raw display
$71$ \( T^{6} + 229 T^{4} + \cdots + 361201 \) Copy content Toggle raw display
$73$ \( T^{6} + 440 T^{4} + \cdots + 2096704 \) Copy content Toggle raw display
$79$ \( (T^{3} + 37 T^{2} + \cdots + 1217)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 369 T^{4} + \cdots + 344569 \) Copy content Toggle raw display
$89$ \( T^{6} + 395 T^{4} + \cdots + 284089 \) Copy content Toggle raw display
$97$ \( T^{6} + 146 T^{4} + \cdots + 32761 \) Copy content Toggle raw display
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