Properties

Label 1155.2.a.t
Level $1155$
Weight $2$
Character orbit 1155.a
Self dual yes
Analytic conductor $9.223$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1155,2,Mod(1,1155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1155, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1155.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1155.a (trivial)

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: yes
Analytic conductor: \(9.22272143346\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

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

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + \beta_1 q^{6} - q^{7} + (\beta_{2} + 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + \beta_1 q^{6} - q^{7} + (\beta_{2} + 1) q^{8} + q^{9} + \beta_1 q^{10} + q^{11} + (\beta_{2} + 1) q^{12} + (\beta_{2} - \beta_1 + 3) q^{13} - \beta_1 q^{14} + q^{15} + ( - \beta_{2} + 2 \beta_1 - 1) q^{16} + ( - \beta_{2} + 2 \beta_1 + 2) q^{17} + \beta_1 q^{18} + ( - \beta_{2} - 2 \beta_1) q^{19} + (\beta_{2} + 1) q^{20} - q^{21} + \beta_1 q^{22} + ( - \beta_{2} - 2) q^{23} + (\beta_{2} + 1) q^{24} + q^{25} + (4 \beta_1 - 2) q^{26} + q^{27} + ( - \beta_{2} - 1) q^{28} + (\beta_1 + 3) q^{29} + \beta_1 q^{30} + (\beta_{2} + \beta_1 - 1) q^{31} + ( - \beta_{2} - 2 \beta_1 + 3) q^{32} + q^{33} + (\beta_{2} + \beta_1 + 5) q^{34} - q^{35} + (\beta_{2} + 1) q^{36} + ( - 3 \beta_{2} - \beta_1 + 3) q^{37} + ( - 3 \beta_{2} - \beta_1 - 7) q^{38} + (\beta_{2} - \beta_1 + 3) q^{39} + (\beta_{2} + 1) q^{40} + ( - 3 \beta_{2} + \beta_1 + 1) q^{41} - \beta_1 q^{42} + (2 \beta_{2} - 3 \beta_1 + 1) q^{43} + (\beta_{2} + 1) q^{44} + q^{45} + ( - \beta_{2} - 3 \beta_1 - 1) q^{46} + ( - \beta_{2} - \beta_1 + 1) q^{47} + ( - \beta_{2} + 2 \beta_1 - 1) q^{48} + q^{49} + \beta_1 q^{50} + ( - \beta_{2} + 2 \beta_1 + 2) q^{51} + (2 \beta_{2} + 6) q^{52} + \beta_{2} q^{53} + \beta_1 q^{54} + q^{55} + ( - \beta_{2} - 1) q^{56} + ( - \beta_{2} - 2 \beta_1) q^{57} + (\beta_{2} + 3 \beta_1 + 3) q^{58} + (3 \beta_1 - 5) q^{59} + (\beta_{2} + 1) q^{60} + (3 \beta_{2} - 2 \beta_1 + 2) q^{61} + (2 \beta_{2} + 4) q^{62} - q^{63} + ( - \beta_{2} - 2 \beta_1 - 5) q^{64} + (\beta_{2} - \beta_1 + 3) q^{65} + \beta_1 q^{66} - 2 \beta_{2} q^{67} + (4 \beta_{2} + 2 \beta_1) q^{68} + ( - \beta_{2} - 2) q^{69} - \beta_1 q^{70} + ( - 3 \beta_{2} + 3 \beta_1 - 7) q^{71} + (\beta_{2} + 1) q^{72} + (2 \beta_1 + 4) q^{73} + ( - 4 \beta_{2} - 6) q^{74} + q^{75} + ( - 2 \beta_{2} - 6 \beta_1 - 6) q^{76} - q^{77} + (4 \beta_1 - 2) q^{78} + ( - \beta_{2} - 7 \beta_1 + 3) q^{79} + ( - \beta_{2} + 2 \beta_1 - 1) q^{80} + q^{81} + ( - 2 \beta_{2} - 2 \beta_1) q^{82} + (3 \beta_{2} - 6 \beta_1 + 4) q^{83} + ( - \beta_{2} - 1) q^{84} + ( - \beta_{2} + 2 \beta_1 + 2) q^{85} + ( - \beta_{2} + 3 \beta_1 - 7) q^{86} + (\beta_1 + 3) q^{87} + (\beta_{2} + 1) q^{88} + (4 \beta_{2} - \beta_1 - 3) q^{89} + \beta_1 q^{90} + ( - \beta_{2} + \beta_1 - 3) q^{91} + ( - 2 \beta_{2} - 2 \beta_1 - 6) q^{92} + (\beta_{2} + \beta_1 - 1) q^{93} + ( - 2 \beta_{2} - 4) q^{94} + ( - \beta_{2} - 2 \beta_1) q^{95} + ( - \beta_{2} - 2 \beta_1 + 3) q^{96} + (6 \beta_{2} - 5 \beta_1 + 5) q^{97} + \beta_1 q^{98} + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} + q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + 3 q^{3} + 3 q^{4} + 3 q^{5} + q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9} + q^{10} + 3 q^{11} + 3 q^{12} + 8 q^{13} - q^{14} + 3 q^{15} - q^{16} + 8 q^{17} + q^{18} - 2 q^{19} + 3 q^{20} - 3 q^{21} + q^{22} - 6 q^{23} + 3 q^{24} + 3 q^{25} - 2 q^{26} + 3 q^{27} - 3 q^{28} + 10 q^{29} + q^{30} - 2 q^{31} + 7 q^{32} + 3 q^{33} + 16 q^{34} - 3 q^{35} + 3 q^{36} + 8 q^{37} - 22 q^{38} + 8 q^{39} + 3 q^{40} + 4 q^{41} - q^{42} + 3 q^{44} + 3 q^{45} - 6 q^{46} + 2 q^{47} - q^{48} + 3 q^{49} + q^{50} + 8 q^{51} + 18 q^{52} + q^{54} + 3 q^{55} - 3 q^{56} - 2 q^{57} + 12 q^{58} - 12 q^{59} + 3 q^{60} + 4 q^{61} + 12 q^{62} - 3 q^{63} - 17 q^{64} + 8 q^{65} + q^{66} + 2 q^{68} - 6 q^{69} - q^{70} - 18 q^{71} + 3 q^{72} + 14 q^{73} - 18 q^{74} + 3 q^{75} - 24 q^{76} - 3 q^{77} - 2 q^{78} + 2 q^{79} - q^{80} + 3 q^{81} - 2 q^{82} + 6 q^{83} - 3 q^{84} + 8 q^{85} - 18 q^{86} + 10 q^{87} + 3 q^{88} - 10 q^{89} + q^{90} - 8 q^{91} - 20 q^{92} - 2 q^{93} - 12 q^{94} - 2 q^{95} + 7 q^{96} + 10 q^{97} + q^{98} + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−1.81361
0.470683
2.34292
−1.81361 1.00000 1.28917 1.00000 −1.81361 −1.00000 1.28917 1.00000 −1.81361
1.2 0.470683 1.00000 −1.77846 1.00000 0.470683 −1.00000 −1.77846 1.00000 0.470683
1.3 2.34292 1.00000 3.48929 1.00000 2.34292 −1.00000 3.48929 1.00000 2.34292
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(-1\)
\(7\) \(1\)
\(11\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1155.2.a.t 3
3.b odd 2 1 3465.2.a.bb 3
5.b even 2 1 5775.2.a.bq 3
7.b odd 2 1 8085.2.a.bl 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.a.t 3 1.a even 1 1 trivial
3465.2.a.bb 3 3.b odd 2 1
5775.2.a.bq 3 5.b even 2 1
8085.2.a.bl 3 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1155))\):

\( T_{2}^{3} - T_{2}^{2} - 4T_{2} + 2 \) Copy content Toggle raw display
\( T_{13}^{3} - 8T_{13}^{2} + 14T_{13} + 4 \) Copy content Toggle raw display
\( T_{17}^{3} - 8T_{17}^{2} + 5T_{17} + 46 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} - T^{2} - 4T + 2 \) Copy content Toggle raw display
$3$ \( (T - 1)^{3} \) Copy content Toggle raw display
$5$ \( (T - 1)^{3} \) Copy content Toggle raw display
$7$ \( (T + 1)^{3} \) Copy content Toggle raw display
$11$ \( (T - 1)^{3} \) Copy content Toggle raw display
$13$ \( T^{3} - 8 T^{2} + 14 T + 4 \) Copy content Toggle raw display
$17$ \( T^{3} - 8 T^{2} + 5 T + 46 \) Copy content Toggle raw display
$19$ \( T^{3} + 2 T^{2} - 31 T + 44 \) Copy content Toggle raw display
$23$ \( T^{3} + 6 T^{2} + 5 T - 8 \) Copy content Toggle raw display
$29$ \( T^{3} - 10 T^{2} + 29 T - 22 \) Copy content Toggle raw display
$31$ \( T^{3} + 2 T^{2} - 14 T - 32 \) Copy content Toggle raw display
$37$ \( T^{3} - 8 T^{2} - 58 T + 292 \) Copy content Toggle raw display
$41$ \( T^{3} - 4 T^{2} - 50 T - 68 \) Copy content Toggle raw display
$43$ \( T^{3} - 43T - 44 \) Copy content Toggle raw display
$47$ \( T^{3} - 2 T^{2} - 14 T + 32 \) Copy content Toggle raw display
$53$ \( T^{3} - 7T + 2 \) Copy content Toggle raw display
$59$ \( T^{3} + 12 T^{2} + 9 T - 76 \) Copy content Toggle raw display
$61$ \( T^{3} - 4 T^{2} - 51 T + 226 \) Copy content Toggle raw display
$67$ \( T^{3} - 28T - 16 \) Copy content Toggle raw display
$71$ \( T^{3} + 18 T^{2} + 42 T - 272 \) Copy content Toggle raw display
$73$ \( T^{3} - 14 T^{2} + 48 T - 16 \) Copy content Toggle raw display
$79$ \( T^{3} - 2 T^{2} - 246 T + 608 \) Copy content Toggle raw display
$83$ \( T^{3} - 6 T^{2} - 135 T - 292 \) Copy content Toggle raw display
$89$ \( T^{3} + 10 T^{2} - 67 T - 2 \) Copy content Toggle raw display
$97$ \( T^{3} - 10 T^{2} - 207 T + 1822 \) Copy content Toggle raw display
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