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Properties

Label 605.2.j.c
Level $605$
Weight $2$
Character orbit 605.j
Analytic conductor $4.831$
Analytic rank $0$
Dimension $8$
CM discriminant -55
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [605,2,Mod(9,605)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(605, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("605.9");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 605 = 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 605.j (of order \(10\), degree \(4\), 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: \(4.83094932229\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.484000000.6
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{6} + 16x^{4} - 66x^{2} + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} + \beta_1) q^{2} + ( - 3 \beta_{5} + 2 \beta_{3} - 2 \beta_{2}) q^{4} + ( - \beta_{5} + \beta_{3} + \beta_{2} + 1) q^{5} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_1) q^{7} + ( - \beta_{7} - 2 \beta_{6}) q^{8} - 3 \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} + \beta_1) q^{2} + ( - 3 \beta_{5} + 2 \beta_{3} - 2 \beta_{2}) q^{4} + ( - \beta_{5} + \beta_{3} + \beta_{2} + 1) q^{5} + (2 \beta_{7} - 2 \beta_{6} + 2 \beta_1) q^{7} + ( - \beta_{7} - 2 \beta_{6}) q^{8} - 3 \beta_{3} q^{9} + ( - \beta_{7} + 2 \beta_{4} - \beta_1) q^{10} + ( - 2 \beta_{7} + 2 \beta_{6} + \cdots - 2 \beta_1) q^{13}+ \cdots + (5 \beta_{7} + 4 \beta_{4} + 5 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 14 q^{4} + 10 q^{5} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 14 q^{4} + 10 q^{5} - 6 q^{9} + 32 q^{14} + 18 q^{16} - 10 q^{20} - 10 q^{25} - 4 q^{26} - 40 q^{31} - 72 q^{34} + 42 q^{36} + 34 q^{49} + 120 q^{56} + 8 q^{59} + 34 q^{64} + 40 q^{70} - 16 q^{71} - 30 q^{80} - 18 q^{81} + 36 q^{86} - 56 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{6} + 16x^{4} - 66x^{2} + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -7\nu^{6} - 37\nu^{4} - 629\nu^{2} - 363 ) / 1991 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -28\nu^{6} - 148\nu^{4} - 525\nu^{2} + 539 ) / 1991 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -28\nu^{7} - 148\nu^{5} - 525\nu^{3} + 539\nu ) / 1991 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 40\nu^{6} - 73\nu^{4} + 750\nu^{2} - 2761 ) / 1991 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 61\nu^{7} + 38\nu^{5} + 646\nu^{3} - 1672\nu ) / 1991 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 68\nu^{7} + 75\nu^{5} + 1275\nu^{3} - 3300\nu ) / 1991 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 4\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{7} - 4\beta_{6} + \beta_{4} + 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -7\beta_{5} - 10\beta_{3} - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{7} - 17\beta_{4} - 7\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 37\beta_{5} - 37\beta_{3} + 75\beta_{2} + 75 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -38\beta_{7} + 75\beta_{6} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/605\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(486\)
\(\chi(n)\) \(-1\) \(\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−1.46782 + 0.476925i
1.46782 − 0.476925i
1.26313 − 1.73855i
−1.26313 + 1.73855i
−1.46782 − 0.476925i
1.46782 + 0.476925i
1.26313 + 1.73855i
−1.26313 − 1.73855i
−2.37499 − 0.771681i 0 3.42705 + 2.48990i 0.690983 + 2.12663i 0 −1.81433 + 2.49721i −3.28216 − 4.51750i 0.927051 − 2.85317i − 5.58394i
9.2 2.37499 + 0.771681i 0 3.42705 + 2.48990i 0.690983 + 2.12663i 0 1.81433 − 2.49721i 3.28216 + 4.51750i 0.927051 − 2.85317i 5.58394i
124.1 −0.780656 − 1.07448i 0 0.0729490 − 0.224514i 1.80902 + 1.31433i 0 −4.08757 − 1.32813i −2.82444 + 0.917716i −2.42705 + 1.76336i − 2.96979i
124.2 0.780656 + 1.07448i 0 0.0729490 − 0.224514i 1.80902 + 1.31433i 0 4.08757 + 1.32813i 2.82444 − 0.917716i −2.42705 + 1.76336i 2.96979i
269.1 −2.37499 + 0.771681i 0 3.42705 − 2.48990i 0.690983 − 2.12663i 0 −1.81433 − 2.49721i −3.28216 + 4.51750i 0.927051 + 2.85317i 5.58394i
269.2 2.37499 − 0.771681i 0 3.42705 − 2.48990i 0.690983 − 2.12663i 0 1.81433 + 2.49721i 3.28216 − 4.51750i 0.927051 + 2.85317i − 5.58394i
444.1 −0.780656 + 1.07448i 0 0.0729490 + 0.224514i 1.80902 − 1.31433i 0 −4.08757 + 1.32813i −2.82444 − 0.917716i −2.42705 − 1.76336i 2.96979i
444.2 0.780656 − 1.07448i 0 0.0729490 + 0.224514i 1.80902 − 1.31433i 0 4.08757 − 1.32813i 2.82444 + 0.917716i −2.42705 − 1.76336i − 2.96979i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
55.d odd 2 1 CM by \(\Q(\sqrt{-55}) \)
5.b even 2 1 inner
11.b odd 2 1 inner
11.c even 5 1 inner
11.d odd 10 1 inner
55.h odd 10 1 inner
55.j even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.j.c 8
5.b even 2 1 inner 605.2.j.c 8
11.b odd 2 1 inner 605.2.j.c 8
11.c even 5 1 605.2.b.b ✓ 4
11.c even 5 2 605.2.j.a 8
11.c even 5 1 inner 605.2.j.c 8
11.d odd 10 1 605.2.b.b ✓ 4
11.d odd 10 2 605.2.j.a 8
11.d odd 10 1 inner 605.2.j.c 8
55.d odd 2 1 CM 605.2.j.c 8
55.h odd 10 1 605.2.b.b ✓ 4
55.h odd 10 2 605.2.j.a 8
55.h odd 10 1 inner 605.2.j.c 8
55.j even 10 1 605.2.b.b ✓ 4
55.j even 10 2 605.2.j.a 8
55.j even 10 1 inner 605.2.j.c 8
55.k odd 20 2 3025.2.a.bb 4
55.l even 20 2 3025.2.a.bb 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.b.b ✓ 4 11.c even 5 1
605.2.b.b ✓ 4 11.d odd 10 1
605.2.b.b ✓ 4 55.h odd 10 1
605.2.b.b ✓ 4 55.j even 10 1
605.2.j.a 8 11.c even 5 2
605.2.j.a 8 11.d odd 10 2
605.2.j.a 8 55.h odd 10 2
605.2.j.a 8 55.j even 10 2
605.2.j.c 8 1.a even 1 1 trivial
605.2.j.c 8 5.b even 2 1 inner
605.2.j.c 8 11.b odd 2 1 inner
605.2.j.c 8 11.c even 5 1 inner
605.2.j.c 8 11.d odd 10 1 inner
605.2.j.c 8 55.d odd 2 1 CM
605.2.j.c 8 55.h odd 10 1 inner
605.2.j.c 8 55.j even 10 1 inner
3025.2.a.bb 4 55.k odd 20 2
3025.2.a.bb 4 55.l even 20 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(605, [\chi])\):

\( T_{2}^{8} - 9T_{2}^{6} + 31T_{2}^{4} + 11T_{2}^{2} + 121 \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 9 T^{6} + \cdots + 121 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 5 T^{3} + 15 T^{2} + \cdots + 25)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} - 24 T^{6} + \cdots + 30976 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 24 T^{6} + \cdots + 30976 \) Copy content Toggle raw display
$17$ \( T^{8} - 104 T^{6} + \cdots + 30976 \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( T^{8} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{4} + 20 T^{3} + \cdots + 6400)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 172 T^{2} + 176)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( T^{8} \) Copy content Toggle raw display
$59$ \( (T^{4} - 4 T^{3} + \cdots + 256)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} + 8 T^{3} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 156 T^{6} + \cdots + 453519616 \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} + 204 T^{6} + \cdots + 30976 \) Copy content Toggle raw display
$89$ \( (T^{2} - 180)^{4} \) Copy content Toggle raw display
$97$ \( T^{8} \) Copy content Toggle raw display
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