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Properties

Label 1100.3.e.a
Level $1100$
Weight $3$
Character orbit 1100.e
Analytic conductor $29.973$
Analytic rank $0$
Dimension $4$
CM discriminant -11
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1100,3,Mod(549,1100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1100.549");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1100.e (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: \(29.9728290796\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{33})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 17x^{2} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 44)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{3} - 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + (\beta_{3} - 6) q^{9} - 11 q^{11} + (4 \beta_{2} - \beta_1) q^{23} + (\beta_{2} + 11 \beta_1) q^{27} + (3 \beta_{3} + 17) q^{31} + 11 \beta_1 q^{33} + ( - 4 \beta_{2} + 13 \beta_1) q^{37} + (5 \beta_{2} + 10 \beta_1) q^{47} - 49 q^{49} + (7 \beta_{2} + 14 \beta_1) q^{53} + ( - 3 \beta_{3} + 55) q^{59} + (8 \beta_{2} - 23 \beta_1) q^{67} + (\beta_{3} - 31) q^{69} + (3 \beta_{3} + 65) q^{71} + ( - 2 \beta_{3} + 107) q^{81} + (9 \beta_{3} - 53) q^{89} + (3 \beta_{2} + 25 \beta_1) q^{93} + (8 \beta_{2} - 35 \beta_1) q^{97} + ( - 11 \beta_{3} + 66) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 22 q^{9} - 44 q^{11} + 74 q^{31} - 196 q^{49} + 214 q^{59} - 122 q^{69} + 266 q^{71} + 424 q^{81} - 194 q^{89} + 242 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 17x^{2} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 5\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{3} + 35\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 5\nu^{2} + 43 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - 3\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 43 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{2} + 7\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(101\) \(177\) \(551\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
549.1
− 3.37228i
− 2.37228i
2.37228i
3.37228i
0 − 5.37228i 0 0 0 0 0 −19.8614 0
549.2 0 − 0.372281i 0 0 0 0 0 8.86141 0
549.3 0 0.372281i 0 0 0 0 0 8.86141 0
549.4 0 5.37228i 0 0 0 0 0 −19.8614 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1100.3.e.a 4
5.b even 2 1 inner 1100.3.e.a 4
5.c odd 4 1 44.3.d.a ✓ 2
5.c odd 4 1 1100.3.f.a 2
11.b odd 2 1 CM 1100.3.e.a 4
15.e even 4 1 396.3.f.a 2
20.e even 4 1 176.3.h.b 2
35.f even 4 1 2156.3.h.a 2
40.i odd 4 1 704.3.h.c 2
40.k even 4 1 704.3.h.f 2
55.d odd 2 1 inner 1100.3.e.a 4
55.e even 4 1 44.3.d.a ✓ 2
55.e even 4 1 1100.3.f.a 2
55.k odd 20 4 484.3.f.b 8
55.l even 20 4 484.3.f.b 8
60.l odd 4 1 1584.3.j.c 2
165.l odd 4 1 396.3.f.a 2
220.i odd 4 1 176.3.h.b 2
385.l odd 4 1 2156.3.h.a 2
440.t even 4 1 704.3.h.c 2
440.w odd 4 1 704.3.h.f 2
660.q even 4 1 1584.3.j.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
44.3.d.a ✓ 2 5.c odd 4 1
44.3.d.a ✓ 2 55.e even 4 1
176.3.h.b 2 20.e even 4 1
176.3.h.b 2 220.i odd 4 1
396.3.f.a 2 15.e even 4 1
396.3.f.a 2 165.l odd 4 1
484.3.f.b 8 55.k odd 20 4
484.3.f.b 8 55.l even 20 4
704.3.h.c 2 40.i odd 4 1
704.3.h.c 2 440.t even 4 1
704.3.h.f 2 40.k even 4 1
704.3.h.f 2 440.w odd 4 1
1100.3.e.a 4 1.a even 1 1 trivial
1100.3.e.a 4 5.b even 2 1 inner
1100.3.e.a 4 11.b odd 2 1 CM
1100.3.e.a 4 55.d odd 2 1 inner
1100.3.f.a 2 5.c odd 4 1
1100.3.f.a 2 55.e even 4 1
1584.3.j.c 2 60.l odd 4 1
1584.3.j.c 2 660.q even 4 1
2156.3.h.a 2 35.f even 4 1
2156.3.h.a 2 385.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 29T_{3}^{2} + 4 \) acting on \(S_{3}^{\mathrm{new}}(1100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 29T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T + 11)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( T^{4} \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 1949 T^{2} + 131044 \) Copy content Toggle raw display
$29$ \( T^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 37 T - 1514)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 7589 T^{2} + 12124324 \) Copy content Toggle raw display
$41$ \( T^{4} \) Copy content Toggle raw display
$43$ \( T^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} + 2500)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 4900)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 107 T + 1006)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} \) Copy content Toggle raw display
$67$ \( T^{4} + 25709 T^{2} + 149866564 \) Copy content Toggle raw display
$71$ \( (T^{2} - 133 T + 2566)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( T^{4} \) Copy content Toggle raw display
$83$ \( T^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} + 97 T - 14354)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 47429 T^{2} + 368716804 \) Copy content Toggle raw display
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