Properties

Label 1100.3.f.f
Level $1100$
Weight $3$
Character orbit 1100.f
Analytic conductor $29.973$
Analytic rank $0$
Dimension $8$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1100,3,Mod(901,1100)]
 
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, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1100.901");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1100 = 2^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1100.f (of order \(2\), degree \(1\), 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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 67x^{6} + 1356x^{4} + 9065x^{2} + 17275 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 5 \)
Twist minimal: no (minimal twist has level 220)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{3} - \beta_{4} q^{7} + ( - \beta_{7} + \beta_1 + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 1) q^{3} - \beta_{4} q^{7} + ( - \beta_{7} + \beta_1 + 4) q^{9} + ( - \beta_{5} + \beta_{2} - \beta_1 + 3) q^{11} + \beta_{6} q^{13} + (\beta_{4} - 2 \beta_{3}) q^{17} + ( - \beta_{4} + \beta_{3}) q^{19} + ( - 2 \beta_{6} - \beta_{5} + \cdots + 2 \beta_{2}) q^{21}+ \cdots + ( - 2 \beta_{6} - 6 \beta_{5} + \cdots - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{3} + 28 q^{9} + 24 q^{11} - 56 q^{23} + 80 q^{27} + 20 q^{31} - 88 q^{33} - 72 q^{37} + 184 q^{47} - 244 q^{49} - 136 q^{53} - 16 q^{59} + 264 q^{67} - 56 q^{69} - 220 q^{71} - 208 q^{77} - 224 q^{81} + 220 q^{89} - 72 q^{91} + 104 q^{93} + 8 q^{97} - 40 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 67x^{6} + 1356x^{4} + 9065x^{2} + 17275 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{6} - 59\nu^{4} - 763\nu^{2} - 420 ) / 605 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{7} - 5\nu^{6} - 129\nu^{5} - 350\nu^{4} - 2692\nu^{3} - 6620\nu^{2} - 25315\nu - 24650 ) / 3025 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 9\nu^{7} + 608\nu^{5} + 12004\nu^{3} + 60155\nu ) / 3025 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} - 57\nu^{5} - 826\nu^{3} - 2020\nu ) / 275 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{6} - 140\nu^{4} - 2648\nu^{2} - 9860 ) / 605 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -23\nu^{7} - 1456\nu^{5} - 25018\nu^{3} - 96835\nu ) / 3025 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 6\nu^{6} + 332\nu^{4} + 4666\nu^{2} + 14070 ) / 605 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{6} + 3\beta_{5} + 2\beta_{4} - 4\beta_{3} - 6\beta_{2} ) / 20 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - \beta_{5} + 8\beta _1 - 34 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 92\beta_{6} - 43\beta_{5} - 82\beta_{4} + 154\beta_{3} + 86\beta_{2} ) / 20 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -51\beta_{7} - 4\beta_{5} - 298\beta _1 + 914 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -3542\beta_{6} + 743\beta_{5} + 3582\beta_{4} - 5004\beta_{3} - 1486\beta_{2} ) / 20 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2246\beta_{7} + 999\beta_{5} + 10268\beta _1 - 28824 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 129942\beta_{6} - 12893\beta_{5} - 145982\beta_{4} + 166104\beta_{3} + 25786\beta_{2} ) / 20 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
901.1
5.98859i
5.98859i
4.54951i
4.54951i
1.79638i
1.79638i
2.68549i
2.68549i
0 −3.65160 0 0 0 9.09184i 0 4.33416 0
901.2 0 −3.65160 0 0 0 9.09184i 0 4.33416 0
901.3 0 −0.712855 0 0 0 7.86633i 0 −8.49184 0
901.4 0 −0.712855 0 0 0 7.86633i 0 −8.49184 0
901.5 0 3.41553 0 0 0 0.558647i 0 2.66584 0
901.6 0 3.41553 0 0 0 0.558647i 0 2.66584 0
901.7 0 4.94892 0 0 0 13.1585i 0 15.4918 0
901.8 0 4.94892 0 0 0 13.1585i 0 15.4918 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 901.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b 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.f.f 8
5.b even 2 1 220.3.f.a 8
5.c odd 4 2 1100.3.e.b 16
11.b odd 2 1 inner 1100.3.f.f 8
15.d odd 2 1 1980.3.b.a 8
20.d odd 2 1 880.3.j.b 8
55.d odd 2 1 220.3.f.a 8
55.e even 4 2 1100.3.e.b 16
165.d even 2 1 1980.3.b.a 8
220.g even 2 1 880.3.j.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
220.3.f.a 8 5.b even 2 1
220.3.f.a 8 55.d odd 2 1
880.3.j.b 8 20.d odd 2 1
880.3.j.b 8 220.g even 2 1
1100.3.e.b 16 5.c odd 4 2
1100.3.e.b 16 55.e even 4 2
1100.3.f.f 8 1.a even 1 1 trivial
1100.3.f.f 8 11.b odd 2 1 inner
1980.3.b.a 8 15.d odd 2 1
1980.3.b.a 8 165.d even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 4T_{3}^{3} - 17T_{3}^{2} + 52T_{3} + 44 \) acting on \(S_{3}^{\mathrm{new}}(1100, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T^{4} - 4 T^{3} - 17 T^{2} + \cdots + 44)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 318 T^{6} + \cdots + 276400 \) Copy content Toggle raw display
$11$ \( T^{8} - 24 T^{7} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( T^{8} + 1052 T^{6} + \cdots + 110560000 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 12073428400 \) Copy content Toggle raw display
$19$ \( T^{8} + 650 T^{6} + \cdots + 110560000 \) Copy content Toggle raw display
$23$ \( (T^{4} + 28 T^{3} + \cdots - 704)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots + 100835142400 \) Copy content Toggle raw display
$31$ \( (T^{4} - 10 T^{3} + \cdots + 1062716)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 36 T^{3} + \cdots - 36620)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 7245660160000 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 840680550400 \) Copy content Toggle raw display
$47$ \( (T^{4} - 92 T^{3} + \cdots - 1600)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 68 T^{3} + \cdots + 181444)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 8 T^{3} + \cdots - 735680)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 75981364960000 \) Copy content Toggle raw display
$67$ \( (T^{4} - 132 T^{3} + \cdots + 10804784)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 110 T^{3} + \cdots - 24870124)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 22293318400 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{4} - 110 T^{3} + \cdots + 44004556)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 4 T^{3} + \cdots + 7804400)^{2} \) Copy content Toggle raw display
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