Properties

Label 1100.3.f.e
Level $1100$
Weight $3$
Character orbit 1100.f
Analytic conductor $29.973$
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] = [1100,3,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: \( x^{8} + 83x^{6} + 1611x^{4} + 7105x^{2} + 400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 5 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + 1) q^{3} + \beta_1 q^{7} + (\beta_{3} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + 1) q^{3} + \beta_1 q^{7} + (\beta_{3} + 2) q^{9} - \beta_{6} q^{11} - \beta_{7} q^{13} + (\beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_1) q^{17} + ( - \beta_{7} - \beta_{6} - \beta_{5} + 2 \beta_{4} + 2 \beta_1) q^{19} + ( - \beta_{7} + \beta_{4} + 2 \beta_1) q^{21} + ( - 2 \beta_{3} + \beta_{2} - 4) q^{23} + ( - \beta_{6} + \beta_{5} + \beta_{3} + \beta_{2}) q^{27} + (3 \beta_{7} - \beta_1) q^{29} + ( - \beta_{6} + \beta_{5} + 2 \beta_{3} + 1) q^{31} + ( - \beta_{7} + \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{33} + ( - 2 \beta_{6} + 2 \beta_{5} + 3 \beta_{3} - \beta_{2} + 2) q^{37} + (\beta_{7} - 2 \beta_{6} - 2 \beta_{5} + \beta_{4} + 3 \beta_1) q^{39} + (3 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} - \beta_{4} - 6 \beta_1) q^{41} + (4 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} - 3 \beta_{4} - 2 \beta_1) q^{43} + ( - 2 \beta_{6} + 2 \beta_{5} - \beta_{3} + 5 \beta_{2}) q^{47} + (\beta_{6} - \beta_{5} + 2 \beta_{3} + 5 \beta_{2}) q^{49} + (2 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} + 4 \beta_{4} + \beta_1) q^{51} + ( - \beta_{6} + \beta_{5} - 5 \beta_{3} - 13) q^{53} + ( - 3 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} + 3 \beta_{4} + 7 \beta_1) q^{57} + (2 \beta_{6} - 2 \beta_{5} + 5 \beta_{3} - 5 \beta_{2} + 20) q^{59} + ( - 2 \beta_{7} - 4 \beta_{4} + 9 \beta_1) q^{61} + ( - \beta_{7} + 5 \beta_{4}) q^{63} + ( - 3 \beta_{6} + 3 \beta_{5} - 4 \beta_{3} - 6 \beta_{2} + 6) q^{67} + (2 \beta_{6} - 2 \beta_{5} - 3 \beta_{3} + 15 \beta_{2} - 28) q^{69} + ( - 3 \beta_{6} + 3 \beta_{5} - \beta_{3} + 5 \beta_{2} - 9) q^{71} + (5 \beta_{6} + 5 \beta_{5} - 3 \beta_1) q^{73} + (5 \beta_{7} + 2 \beta_{6} + 6 \beta_{5} - \beta_{4} + \beta_{3} - 5 \beta_{2} - \beta_1 + 18) q^{77} + ( - 5 \beta_{7} - 4 \beta_{6} - 4 \beta_{5} - \beta_{4} - 5 \beta_1) q^{79} + ( - 2 \beta_{6} + 2 \beta_{5} - 5 \beta_{3} - 5 \beta_{2} - 15) q^{81} + ( - 5 \beta_{7} - 3 \beta_{6} - 3 \beta_{5} + 2 \beta_{4} + 15 \beta_1) q^{83} + ( - 2 \beta_{7} + 6 \beta_{6} + 6 \beta_{5} - 4 \beta_{4} - 11 \beta_1) q^{87} + (3 \beta_{6} - 3 \beta_{5} - 9 \beta_{3} - 20 \beta_{2} - 6) q^{89} + ( - 3 \beta_{6} + 3 \beta_{5} - \beta_{3} + 10 \beta_{2} + 7) q^{91} + ( - 3 \beta_{6} + 3 \beta_{5} + 6 \beta_{3} - 11 \beta_{2} + 21) q^{93} + ( - 6 \beta_{6} + 6 \beta_{5} - 2 \beta_{3} + \beta_{2} - 42) q^{97} + (4 \beta_{7} - 4 \beta_{5} - 3 \beta_{4} + 3 \beta_{3} - 15 \beta_{2} - 3 \beta_1 + 10) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} + 16 q^{9} + 3 q^{11} - 28 q^{23} + 10 q^{27} + 14 q^{31} + 31 q^{33} + 24 q^{37} + 32 q^{47} + 14 q^{49} - 98 q^{53} + 128 q^{59} + 42 q^{67} - 176 q^{69} - 34 q^{71} + 136 q^{77} - 128 q^{81} - 146 q^{89} + 114 q^{91} + 142 q^{93} - 296 q^{97} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 83x^{6} + 1611x^{4} + 7105x^{2} + 400 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 69\nu^{3} + 790\nu ) / 110 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 69\nu^{2} + 570 ) / 110 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 74\nu^{4} + 1025\nu^{2} + 1420 ) / 220 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 83\nu^{5} + 1591\nu^{3} + 5725\nu ) / 220 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -3\nu^{7} - \nu^{6} - 246\nu^{5} - 84\nu^{4} - 4621\nu^{3} - 1495\nu^{2} - 19150\nu - 1840 ) / 440 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{7} + \nu^{6} - 246\nu^{5} + 84\nu^{4} - 4621\nu^{3} + 1495\nu^{2} - 19150\nu + 1840 ) / 440 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{7} + 167\nu^{5} + 3306\nu^{3} + 15485\nu ) / 220 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + 2\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + \beta_{5} + \beta_{3} + 5\beta_{2} - 24 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 79\beta_{7} + 59\beta_{6} + 59\beta_{5} + 19\beta_{4} - 128\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 69\beta_{6} - 69\beta_{5} - 69\beta_{3} - 235\beta_{2} + 1086 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -4661\beta_{7} - 3281\beta_{6} - 3281\beta_{5} - 521\beta_{4} + 7802\beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -4081\beta_{6} + 4081\beta_{5} + 4301\beta_{3} + 12265\beta_{2} - 57184 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 266899\beta_{7} + 184179\beta_{6} + 184179\beta_{5} + 19839\beta_{4} - 455368\beta_1 ) / 5 \) 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
0.238817i
0.238817i
2.49129i
2.49129i
7.54086i
7.54086i
4.45778i
4.45778i
0 −4.14607 0 0 0 1.70661i 0 8.18991 0
901.2 0 −4.14607 0 0 0 1.70661i 0 8.18991 0
901.3 0 −0.638826 0 0 0 9.06537i 0 −8.59190 0
901.4 0 −0.638826 0 0 0 9.06537i 0 −8.59190 0
901.5 0 2.09157 0 0 0 6.85033i 0 −4.62533 0
901.6 0 2.09157 0 0 0 6.85033i 0 −4.62533 0
901.7 0 4.69333 0 0 0 7.54848i 0 13.0273 0
901.8 0 4.69333 0 0 0 7.54848i 0 13.0273 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.e yes 8
5.b even 2 1 1100.3.f.c 8
5.c odd 4 2 1100.3.e.c 16
11.b odd 2 1 inner 1100.3.f.e yes 8
55.d odd 2 1 1100.3.f.c 8
55.e even 4 2 1100.3.e.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1100.3.e.c 16 5.c odd 4 2
1100.3.e.c 16 55.e even 4 2
1100.3.f.c 8 5.b even 2 1
1100.3.f.c 8 55.d odd 2 1
1100.3.f.e yes 8 1.a even 1 1 trivial
1100.3.f.e yes 8 11.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} - 2T_{3}^{3} - 20T_{3}^{2} + 29T_{3} + 26 \) 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
$3$ \( (T^{4} - 2 T^{3} - 20 T^{2} + 29 T + 26)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 189 T^{6} + 11755 T^{4} + \cdots + 640000 \) Copy content Toggle raw display
$11$ \( T^{8} - 3 T^{7} - 30 T^{6} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( T^{8} + 461 T^{6} + 59455 T^{4} + \cdots + 1562500 \) Copy content Toggle raw display
$17$ \( T^{8} + 1854 T^{6} + \cdots + 2560000 \) Copy content Toggle raw display
$19$ \( T^{8} + 2660 T^{6} + \cdots + 71355765625 \) Copy content Toggle raw display
$23$ \( (T^{4} + 14 T^{3} - 618 T^{2} + \cdots + 51322)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 3996 T^{6} + \cdots + 125139062500 \) Copy content Toggle raw display
$31$ \( (T^{4} - 7 T^{3} - 1135 T^{2} + 6187 T - 46)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} - 12 T^{3} - 3587 T^{2} + \cdots + 17200)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 7715 T^{6} + \cdots + 175561000000 \) Copy content Toggle raw display
$43$ \( T^{8} + 9721 T^{6} + \cdots + 207571360000 \) Copy content Toggle raw display
$47$ \( (T^{4} - 16 T^{3} - 2465 T^{2} + \cdots + 54500)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 49 T^{3} - 3557 T^{2} + \cdots - 115700)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 64 T^{3} - 5181 T^{2} + \cdots - 8018420)^{2} \) Copy content Toggle raw display
$61$ \( T^{8} + 27421 T^{6} + \cdots + 20675209000000 \) Copy content Toggle raw display
$67$ \( (T^{4} - 21 T^{3} - 7844 T^{2} + \cdots + 3185024)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 17 T^{3} - 4890 T^{2} + \cdots - 250696)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 20096 T^{6} + \cdots + 50809096802500 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 121475666560000 \) Copy content Toggle raw display
$83$ \( T^{8} + 34291 T^{6} + \cdots + 57826519140625 \) Copy content Toggle raw display
$89$ \( (T^{4} + 73 T^{3} - 18920 T^{2} + \cdots - 5154671)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 148 T^{3} - 10902 T^{2} + \cdots + 2535410)^{2} \) Copy content Toggle raw display
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