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Properties

Label 363.2.e.m
Level $363$
Weight $2$
Character orbit 363.e
Analytic conductor $2.899$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [363,2,Mod(124,363)]
 
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([0, 8]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("363.124");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 363 = 3 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 363.e (of order \(5\), 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 \(\Q(\zeta_{5})\)
Coefficient field: 8.0.324000000.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} - \beta_{6} q^{3} + \beta_{4} q^{4} + (3 \beta_{6} + 3 \beta_{4} + 3 \beta_{2} + 3) q^{5} - \beta_{3} q^{6} + (2 \beta_{7} + 2 \beta_{5} + \cdots + 2 \beta_1) q^{7}+ \cdots + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} - \beta_{6} q^{3} + \beta_{4} q^{4} + (3 \beta_{6} + 3 \beta_{4} + 3 \beta_{2} + 3) q^{5} - \beta_{3} q^{6} + (2 \beta_{7} + 2 \beta_{5} + \cdots + 2 \beta_1) q^{7}+ \cdots + 5 \beta_{5} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} - 2 q^{4} + 6 q^{5} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{3} - 2 q^{4} + 6 q^{5} - 2 q^{9} - 8 q^{12} + 12 q^{14} - 6 q^{15} + 10 q^{16} + 6 q^{20} - 48 q^{23} - 8 q^{25} + 6 q^{26} + 2 q^{27} - 8 q^{31} + 24 q^{34} - 2 q^{36} + 22 q^{37} - 24 q^{38} - 12 q^{42} - 24 q^{45} - 10 q^{48} - 10 q^{49} + 18 q^{53} + 48 q^{56} + 6 q^{58} + 12 q^{59} - 6 q^{60} - 2 q^{64} - 16 q^{67} - 12 q^{69} - 36 q^{70} + 12 q^{71} + 8 q^{75} + 24 q^{78} - 30 q^{80} - 2 q^{81} + 6 q^{82} - 12 q^{86} + 72 q^{89} - 12 q^{91} + 12 q^{92} + 8 q^{93} + 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 3x^{6} + 9x^{4} + 27x^{2} + 81 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} ) / 27 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 9\beta_{4} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{5} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 27\beta_{6} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 27\beta_{7} \) Copy content Toggle raw display

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

Character values

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

\(n\) \(122\) \(244\)
\(\chi(n)\) \(1\) \(\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
124.1
0.535233 + 1.64728i
−0.535233 − 1.64728i
1.40126 − 1.01807i
−1.40126 + 1.01807i
1.40126 + 1.01807i
−1.40126 − 1.01807i
0.535233 − 1.64728i
−0.535233 + 1.64728i
−1.40126 + 1.01807i −0.309017 − 0.951057i 0.309017 − 0.951057i 2.42705 + 1.76336i 1.40126 + 1.01807i −1.07047 + 3.29456i −0.535233 − 1.64728i −0.809017 + 0.587785i −5.19615
124.2 1.40126 − 1.01807i −0.309017 − 0.951057i 0.309017 − 0.951057i 2.42705 + 1.76336i −1.40126 − 1.01807i 1.07047 − 3.29456i 0.535233 + 1.64728i −0.809017 + 0.587785i 5.19615
130.1 −0.535233 + 1.64728i 0.809017 − 0.587785i −0.809017 − 0.587785i −0.927051 − 2.85317i 0.535233 + 1.64728i −2.80252 − 2.03615i −1.40126 + 1.01807i 0.309017 − 0.951057i 5.19615
130.2 0.535233 − 1.64728i 0.809017 − 0.587785i −0.809017 − 0.587785i −0.927051 − 2.85317i −0.535233 − 1.64728i 2.80252 + 2.03615i 1.40126 − 1.01807i 0.309017 − 0.951057i −5.19615
148.1 −0.535233 − 1.64728i 0.809017 + 0.587785i −0.809017 + 0.587785i −0.927051 + 2.85317i 0.535233 − 1.64728i −2.80252 + 2.03615i −1.40126 − 1.01807i 0.309017 + 0.951057i 5.19615
148.2 0.535233 + 1.64728i 0.809017 + 0.587785i −0.809017 + 0.587785i −0.927051 + 2.85317i −0.535233 + 1.64728i 2.80252 − 2.03615i 1.40126 + 1.01807i 0.309017 + 0.951057i −5.19615
202.1 −1.40126 − 1.01807i −0.309017 + 0.951057i 0.309017 + 0.951057i 2.42705 − 1.76336i 1.40126 − 1.01807i −1.07047 − 3.29456i −0.535233 + 1.64728i −0.809017 − 0.587785i −5.19615
202.2 1.40126 + 1.01807i −0.309017 + 0.951057i 0.309017 + 0.951057i 2.42705 − 1.76336i −1.40126 + 1.01807i 1.07047 + 3.29456i 0.535233 − 1.64728i −0.809017 − 0.587785i 5.19615
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 124.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner
11.c even 5 3 inner
11.d 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.e.m 8
11.b odd 2 1 inner 363.2.e.m 8
11.c even 5 1 363.2.a.f ✓ 2
11.c even 5 3 inner 363.2.e.m 8
11.d odd 10 1 363.2.a.f ✓ 2
11.d odd 10 3 inner 363.2.e.m 8
33.f even 10 1 1089.2.a.o 2
33.h odd 10 1 1089.2.a.o 2
44.g even 10 1 5808.2.a.ca 2
44.h odd 10 1 5808.2.a.ca 2
55.h odd 10 1 9075.2.a.bo 2
55.j even 10 1 9075.2.a.bo 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
363.2.a.f ✓ 2 11.c even 5 1
363.2.a.f ✓ 2 11.d odd 10 1
363.2.e.m 8 1.a even 1 1 trivial
363.2.e.m 8 11.b odd 2 1 inner
363.2.e.m 8 11.c even 5 3 inner
363.2.e.m 8 11.d odd 10 3 inner
1089.2.a.o 2 33.f even 10 1
1089.2.a.o 2 33.h odd 10 1
5808.2.a.ca 2 44.g even 10 1
5808.2.a.ca 2 44.h odd 10 1
9075.2.a.bo 2 55.h odd 10 1
9075.2.a.bo 2 55.j even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 3T_{2}^{6} + 9T_{2}^{4} + 27T_{2}^{2} + 81 \) acting on \(S_{2}^{\mathrm{new}}(363, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 3 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( (T^{4} - T^{3} + T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} - 3 T^{3} + 9 T^{2} + \cdots + 81)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 12 T^{6} + \cdots + 20736 \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 3 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$17$ \( T^{8} + 3 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( T^{8} + 48 T^{6} + \cdots + 5308416 \) Copy content Toggle raw display
$23$ \( (T + 6)^{8} \) Copy content Toggle raw display
$29$ \( T^{8} + 3 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$31$ \( (T^{4} + 4 T^{3} + \cdots + 256)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 11 T^{3} + \cdots + 14641)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 3 T^{6} + \cdots + 81 \) Copy content Toggle raw display
$43$ \( (T^{2} - 12)^{4} \) Copy content Toggle raw display
$47$ \( T^{8} \) Copy content Toggle raw display
$53$ \( (T^{4} - 9 T^{3} + \cdots + 6561)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 6 T^{3} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} \) Copy content Toggle raw display
$67$ \( (T + 2)^{8} \) Copy content Toggle raw display
$71$ \( (T^{4} - 6 T^{3} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 48 T^{6} + \cdots + 5308416 \) Copy content Toggle raw display
$79$ \( T^{8} \) Copy content Toggle raw display
$83$ \( T^{8} \) Copy content Toggle raw display
$89$ \( (T - 9)^{8} \) Copy content Toggle raw display
$97$ \( (T^{4} - 7 T^{3} + \cdots + 2401)^{2} \) Copy content Toggle raw display
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