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Properties

Label 1100.2.n.c
Level $1100$
Weight $2$
Character orbit 1100.n
Analytic conductor $8.784$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1100,2,Mod(201,1100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1100, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 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("1100.201");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1100.n (of order \(5\), degree \(4\), 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: \(8.78354422234\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: 8.0.26265625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 2x^{6} + x^{4} + 8x^{2} - 24x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 220)
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.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_1) q^{3} + (2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} + \cdots + 1) q^{7}+ \cdots + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + \beta_1) q^{3} + (2 \beta_{7} - 2 \beta_{5} + 2 \beta_{3} + \cdots + 1) q^{7}+ \cdots + ( - \beta_{7} - 6 \beta_{6} + 2 \beta_{5} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{3} - q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{3} - q^{7} + 3 q^{9} + 5 q^{11} - 6 q^{13} + 13 q^{17} - 7 q^{19} + 28 q^{21} + 22 q^{23} - 2 q^{27} + 3 q^{29} - 2 q^{31} + 15 q^{33} - 16 q^{37} - 17 q^{39} - 12 q^{41} - 10 q^{43} + 18 q^{47} + 21 q^{49} + 21 q^{51} + 23 q^{53} + q^{57} - 9 q^{59} - 34 q^{61} + 29 q^{63} - 26 q^{67} - q^{69} - 26 q^{71} - q^{73} + 10 q^{77} + 19 q^{79} + 12 q^{81} + 7 q^{83} - 94 q^{87} - 8 q^{89} - 18 q^{91} - 49 q^{93} + 24 q^{97} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 2x^{6} + x^{4} + 8x^{2} - 24x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - \nu^{6} + \nu^{3} + 2\nu^{2} + 4\nu - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 3\nu^{6} - 2\nu^{5} - \nu^{3} - 8\nu + 24 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + \nu^{6} - \nu^{3} + 2\nu^{2} - 8\nu + 8 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -7\nu^{7} + 9\nu^{6} + 2\nu^{5} + 4\nu^{4} + \nu^{3} - 4\nu^{2} - 60\nu + 64 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -7\nu^{7} + 13\nu^{6} + 2\nu^{5} - 7\nu^{3} - 8\nu^{2} - 64\nu + 96 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6\nu^{7} - 11\nu^{6} - \nu^{5} - 2\nu^{4} + 6\nu^{3} + 7\nu^{2} + 56\nu - 80 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} + 2\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{7} - 3\beta_{6} - \beta_{3} + 2\beta_{2} + 2\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{6} - 3\beta_{3} + 4\beta_{2} + 3\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{6} - \beta_{4} + \beta_{3} + 6\beta_{2} + 4\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{7} - \beta_{5} - 2\beta_{4} + 9\beta_{2} - 2\beta _1 + 1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1100\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\) \(551\)
\(\chi(n)\) \(-1 + \beta_{2} + \beta_{5} + \beta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
201.1
−1.21700 − 0.720348i
1.40799 + 0.132563i
−1.21700 + 0.720348i
1.40799 − 0.132563i
−0.0272949 − 1.41395i
1.33631 + 0.462894i
−0.0272949 + 1.41395i
1.33631 − 0.462894i
0 −0.655837 − 2.01846i 0 0 0 −0.0946704 + 0.291365i 0 −1.21700 + 0.884205i 0
201.2 0 0.346820 + 1.06740i 0 0 0 −0.714347 + 2.19853i 0 1.40799 − 1.02296i 0
301.1 0 −0.655837 + 2.01846i 0 0 0 −0.0946704 − 0.291365i 0 −1.21700 − 0.884205i 0
301.2 0 0.346820 − 1.06740i 0 0 0 −0.714347 − 2.19853i 0 1.40799 + 1.02296i 0
401.1 0 −1.38048 − 1.00297i 0 0 0 −2.73366 + 1.98612i 0 −0.0272949 − 0.0840051i 0
401.2 0 2.18949 + 1.59076i 0 0 0 3.04267 − 2.21063i 0 1.33631 + 4.11275i 0
801.1 0 −1.38048 + 1.00297i 0 0 0 −2.73366 − 1.98612i 0 −0.0272949 + 0.0840051i 0
801.2 0 2.18949 − 1.59076i 0 0 0 3.04267 + 2.21063i 0 1.33631 − 4.11275i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 201.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1100.2.n.c 8
5.b even 2 1 220.2.m.a ✓ 8
5.c odd 4 2 1100.2.cb.c 16
11.c even 5 1 inner 1100.2.n.c 8
15.d odd 2 1 1980.2.z.c 8
20.d odd 2 1 880.2.bo.f 8
55.h odd 10 1 2420.2.a.m 4
55.j even 10 1 220.2.m.a ✓ 8
55.j even 10 1 2420.2.a.n 4
55.k odd 20 2 1100.2.cb.c 16
165.o odd 10 1 1980.2.z.c 8
220.n odd 10 1 880.2.bo.f 8
220.n odd 10 1 9680.2.a.ck 4
220.o even 10 1 9680.2.a.cl 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.2.m.a ✓ 8 5.b even 2 1
220.2.m.a ✓ 8 55.j even 10 1
880.2.bo.f 8 20.d odd 2 1
880.2.bo.f 8 220.n odd 10 1
1100.2.n.c 8 1.a even 1 1 trivial
1100.2.n.c 8 11.c even 5 1 inner
1100.2.cb.c 16 5.c odd 4 2
1100.2.cb.c 16 55.k odd 20 2
1980.2.z.c 8 15.d odd 2 1
1980.2.z.c 8 165.o odd 10 1
2420.2.a.m 4 55.h odd 10 1
2420.2.a.n 4 55.j even 10 1
9680.2.a.ck 4 220.n odd 10 1
9680.2.a.cl 4 220.o even 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - T_{3}^{7} + 2T_{3}^{6} - 3T_{3}^{5} + 25T_{3}^{4} + 43T_{3}^{3} + 82T_{3}^{2} + 11T_{3} + 121 \) acting on \(S_{2}^{\mathrm{new}}(1100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - T^{7} + \cdots + 121 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + T^{7} + \cdots + 81 \) Copy content Toggle raw display
$11$ \( T^{8} - 5 T^{7} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} + 6 T^{7} + \cdots + 7921 \) Copy content Toggle raw display
$17$ \( T^{8} - 13 T^{7} + \cdots + 121 \) Copy content Toggle raw display
$19$ \( T^{8} + 7 T^{7} + \cdots + 1 \) Copy content Toggle raw display
$23$ \( (T^{4} - 11 T^{3} + \cdots - 99)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} - 3 T^{7} + \cdots + 707281 \) Copy content Toggle raw display
$31$ \( T^{8} + 2 T^{7} + \cdots + 2313441 \) Copy content Toggle raw display
$37$ \( T^{8} + 16 T^{7} + \cdots + 1038361 \) Copy content Toggle raw display
$41$ \( T^{8} + 12 T^{7} + \cdots + 6561 \) Copy content Toggle raw display
$43$ \( (T^{4} + 5 T^{3} + \cdots - 145)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 18 T^{7} + \cdots + 10201 \) Copy content Toggle raw display
$53$ \( T^{8} - 23 T^{7} + \cdots + 58081 \) Copy content Toggle raw display
$59$ \( T^{8} + 9 T^{7} + \cdots + 1185921 \) Copy content Toggle raw display
$61$ \( T^{8} + 34 T^{7} + \cdots + 273207841 \) Copy content Toggle raw display
$67$ \( (T^{4} + 13 T^{3} + \cdots + 881)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 26 T^{7} + \cdots + 43546801 \) Copy content Toggle raw display
$73$ \( T^{8} + T^{7} + \cdots + 641601 \) Copy content Toggle raw display
$79$ \( T^{8} - 19 T^{7} + \cdots + 2128681 \) Copy content Toggle raw display
$83$ \( T^{8} - 7 T^{7} + \cdots + 5948721 \) Copy content Toggle raw display
$89$ \( (T^{4} + 4 T^{3} + \cdots + 881)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 24 T^{7} + \cdots + 962361 \) Copy content Toggle raw display
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