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Properties

Label 363.2.f.c
Level $363$
Weight $2$
Character orbit 363.f
Analytic conductor $2.899$
Analytic rank $0$
Dimension $8$
CM discriminant -11
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [363,2,Mod(161,363)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(363, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([5, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.161");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 363.f (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: \(2.89856959337\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{10})\)
Coefficient field: 8.0.228765625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 2x^{6} + 5x^{5} + x^{4} + 15x^{3} - 18x^{2} - 27x + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - 2 \beta_{7} q^{4} + ( - 2 \beta_{5} - \beta_{4}) q^{5} + (2 \beta_{7} + \beta_{6} - \beta_{5} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - 2 \beta_{7} q^{4} + ( - 2 \beta_{5} - \beta_{4}) q^{5} + (2 \beta_{7} + \beta_{6} - \beta_{5} + \cdots + 2) q^{9}+ \cdots + ( - 17 \beta_{7} + 17 \beta_{4} + \cdots - 17) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{3} + 4 q^{4} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - q^{3} + 4 q^{4} + 5 q^{9} - 8 q^{12} - 11 q^{15} - 8 q^{16} + 12 q^{25} + 8 q^{27} - 10 q^{31} - 10 q^{36} + 14 q^{37} + 44 q^{45} - 4 q^{48} - 14 q^{49} + 22 q^{60} + 16 q^{64} - 104 q^{67} - 11 q^{69} + 6 q^{75} - 7 q^{81} - 5 q^{93} - 34 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 2x^{6} + 5x^{5} + x^{4} + 15x^{3} - 18x^{2} - 27x + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{7} + \nu^{6} + 2\nu^{5} - 5\nu^{4} - \nu^{3} - 15\nu^{2} + 18\nu + 27 ) / 27 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 16\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{7} - 4\nu^{6} + 10\nu^{5} + 2\nu^{4} - 5\nu^{3} - 36\nu^{2} - 54\nu + 162 ) / 27 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} - 2\nu^{6} + 5\nu^{5} + \nu^{4} + 2\nu^{3} - 18\nu^{2} - 27\nu + 81 ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{5} + 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{7} + 5\nu^{6} + 10\nu^{5} + 2\nu^{4} - 5\nu^{3} - 75\nu^{2} + 90\nu + 135 ) / 27 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{7} + \beta_{6} - \beta_{5} - 3\beta_{4} - 3\beta_{3} - \beta_{2} + \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} - 3\beta_{4} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - 5\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{6} - 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{3} - 16\beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -35\beta_{7} - 13\beta_{6} + 13\beta_{5} + 48\beta_{4} + 48\beta_{3} + 13\beta_{2} - 13\beta _1 - 35 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/363\mathbb{Z}\right)^\times\).

\(n\) \(122\) \(244\)
\(\chi(n)\) \(-1\) \(-\beta_{7}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
1.42264 + 0.987975i
−1.73166 − 0.0369185i
1.37924 − 1.04771i
−0.570223 + 1.63550i
1.37924 + 1.04771i
−0.570223 − 1.63550i
1.42264 − 0.987975i
−1.73166 + 0.0369185i
0 −1.42264 − 0.987975i −0.618034 − 1.90211i −1.94946 − 2.68321i 0 0 0 1.04781 + 2.81107i 0
161.2 0 1.73166 + 0.0369185i −0.618034 − 1.90211i 1.94946 + 2.68321i 0 0 0 2.99727 + 0.127860i 0
215.1 0 −1.37924 + 1.04771i 1.61803 − 1.17557i 3.15430 + 1.02489i 0 0 0 0.804606 − 2.89009i 0
215.2 0 0.570223 − 1.63550i 1.61803 − 1.17557i −3.15430 − 1.02489i 0 0 0 −2.34969 − 1.86519i 0
233.1 0 −1.37924 − 1.04771i 1.61803 + 1.17557i 3.15430 − 1.02489i 0 0 0 0.804606 + 2.89009i 0
233.2 0 0.570223 + 1.63550i 1.61803 + 1.17557i −3.15430 + 1.02489i 0 0 0 −2.34969 + 1.86519i 0
239.1 0 −1.42264 + 0.987975i −0.618034 + 1.90211i −1.94946 + 2.68321i 0 0 0 1.04781 − 2.81107i 0
239.2 0 1.73166 − 0.0369185i −0.618034 + 1.90211i 1.94946 − 2.68321i 0 0 0 2.99727 − 0.127860i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 CM by \(\Q(\sqrt{-11}) \)
3.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
33.d even 2 1 inner
33.f even 10 3 inner
33.h odd 10 3 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 363.2.f.c 8
3.b odd 2 1 inner 363.2.f.c 8
11.b odd 2 1 CM 363.2.f.c 8
11.c even 5 1 33.2.d.a ✓ 2
11.c even 5 3 inner 363.2.f.c 8
11.d odd 10 1 33.2.d.a ✓ 2
11.d odd 10 3 inner 363.2.f.c 8
33.d even 2 1 inner 363.2.f.c 8
33.f even 10 1 33.2.d.a ✓ 2
33.f even 10 3 inner 363.2.f.c 8
33.h odd 10 1 33.2.d.a ✓ 2
33.h odd 10 3 inner 363.2.f.c 8
44.g even 10 1 528.2.b.a 2
44.h odd 10 1 528.2.b.a 2
55.h odd 10 1 825.2.f.a 2
55.j even 10 1 825.2.f.a 2
55.k odd 20 2 825.2.d.a 4
55.l even 20 2 825.2.d.a 4
88.k even 10 1 2112.2.b.f 2
88.l odd 10 1 2112.2.b.f 2
88.o even 10 1 2112.2.b.e 2
88.p odd 10 1 2112.2.b.e 2
99.m even 15 2 891.2.g.a 4
99.n odd 30 2 891.2.g.a 4
99.o odd 30 2 891.2.g.a 4
99.p even 30 2 891.2.g.a 4
132.n odd 10 1 528.2.b.a 2
132.o even 10 1 528.2.b.a 2
165.o odd 10 1 825.2.f.a 2
165.r even 10 1 825.2.f.a 2
165.u odd 20 2 825.2.d.a 4
165.v even 20 2 825.2.d.a 4
264.r odd 10 1 2112.2.b.f 2
264.t odd 10 1 2112.2.b.e 2
264.u even 10 1 2112.2.b.e 2
264.w even 10 1 2112.2.b.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.2.d.a ✓ 2 11.c even 5 1
33.2.d.a ✓ 2 11.d odd 10 1
33.2.d.a ✓ 2 33.f even 10 1
33.2.d.a ✓ 2 33.h odd 10 1
363.2.f.c 8 1.a even 1 1 trivial
363.2.f.c 8 3.b odd 2 1 inner
363.2.f.c 8 11.b odd 2 1 CM
363.2.f.c 8 11.c even 5 3 inner
363.2.f.c 8 11.d odd 10 3 inner
363.2.f.c 8 33.d even 2 1 inner
363.2.f.c 8 33.f even 10 3 inner
363.2.f.c 8 33.h odd 10 3 inner
528.2.b.a 2 44.g even 10 1
528.2.b.a 2 44.h odd 10 1
528.2.b.a 2 132.n odd 10 1
528.2.b.a 2 132.o even 10 1
825.2.d.a 4 55.k odd 20 2
825.2.d.a 4 55.l even 20 2
825.2.d.a 4 165.u odd 20 2
825.2.d.a 4 165.v even 20 2
825.2.f.a 2 55.h odd 10 1
825.2.f.a 2 55.j even 10 1
825.2.f.a 2 165.o odd 10 1
825.2.f.a 2 165.r even 10 1
891.2.g.a 4 99.m even 15 2
891.2.g.a 4 99.n odd 30 2
891.2.g.a 4 99.o odd 30 2
891.2.g.a 4 99.p even 30 2
2112.2.b.e 2 88.o even 10 1
2112.2.b.e 2 88.p odd 10 1
2112.2.b.e 2 264.t odd 10 1
2112.2.b.e 2 264.u even 10 1
2112.2.b.f 2 88.k even 10 1
2112.2.b.f 2 88.l odd 10 1
2112.2.b.f 2 264.r odd 10 1
2112.2.b.f 2 264.w even 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(363, [\chi])\):

\( T_{2} \) Copy content Toggle raw display
\( T_{5}^{8} - 11T_{5}^{6} + 121T_{5}^{4} - 1331T_{5}^{2} + 14641 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + T^{7} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{8} - 11 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( T^{8} \) Copy content Toggle raw display
$19$ \( T^{8} \) Copy content Toggle raw display
$23$ \( (T^{2} + 11)^{4} \) Copy content Toggle raw display
$29$ \( T^{8} \) Copy content Toggle raw display
$31$ \( (T^{4} + 5 T^{3} + \cdots + 625)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 7 T^{3} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} \) Copy content Toggle raw display
$43$ \( T^{8} \) Copy content Toggle raw display
$47$ \( T^{8} - 44 T^{6} + \cdots + 3748096 \) Copy content Toggle raw display
$53$ \( T^{8} - 176 T^{6} + \cdots + 959512576 \) Copy content Toggle raw display
$59$ \( T^{8} - 11 T^{6} + \cdots + 14641 \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T + 13)^{8} \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 5719140625 \) Copy content Toggle raw display
$73$ \( T^{8} \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( (T^{2} + 275)^{4} \) Copy content Toggle raw display
$97$ \( (T^{4} + 17 T^{3} + \cdots + 83521)^{2} \) Copy content Toggle raw display
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