Properties

Label 1155.2.a.p
Level $1155$
Weight $2$
Character orbit 1155.a
Self dual yes
Analytic conductor $9.223$
Analytic rank $0$
Dimension $2$
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: \(2\)
Coefficient field: \(\Q(\sqrt{2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) 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 \(\beta = \sqrt{2}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta q^{2} - q^{3} - q^{5} - \beta q^{6} - q^{7} - 2 \beta q^{8} + q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} - q^{3} - q^{5} - \beta q^{6} - q^{7} - 2 \beta q^{8} + q^{9} - \beta q^{10} + q^{11} + (\beta + 2) q^{13} - \beta q^{14} + q^{15} - 4 q^{16} + (2 \beta + 1) q^{17} + \beta q^{18} + (2 \beta - 1) q^{19} + q^{21} + \beta q^{22} + (2 \beta - 1) q^{23} + 2 \beta q^{24} + q^{25} + (2 \beta + 2) q^{26} - q^{27} + (3 \beta + 3) q^{29} + \beta q^{30} + (\beta + 2) q^{31} - q^{33} + (\beta + 4) q^{34} + q^{35} + (3 \beta - 2) q^{37} + ( - \beta + 4) q^{38} + ( - \beta - 2) q^{39} + 2 \beta q^{40} + ( - 5 \beta + 4) q^{41} + \beta q^{42} + (\beta - 5) q^{43} - q^{45} + ( - \beta + 4) q^{46} + (3 \beta + 6) q^{47} + 4 q^{48} + q^{49} + \beta q^{50} + ( - 2 \beta - 1) q^{51} + ( - 4 \beta + 3) q^{53} - \beta q^{54} - q^{55} + 2 \beta q^{56} + ( - 2 \beta + 1) q^{57} + (3 \beta + 6) q^{58} + ( - \beta + 13) q^{59} + ( - 4 \beta + 5) q^{61} + (2 \beta + 2) q^{62} - q^{63} + 8 q^{64} + ( - \beta - 2) q^{65} - \beta q^{66} - 6 q^{67} + ( - 2 \beta + 1) q^{69} + \beta q^{70} + ( - 5 \beta - 4) q^{71} - 2 \beta q^{72} + 6 \beta q^{73} + ( - 2 \beta + 6) q^{74} - q^{75} - q^{77} + ( - 2 \beta - 2) q^{78} + (3 \beta - 4) q^{79} + 4 q^{80} + q^{81} + (4 \beta - 10) q^{82} + ( - 10 \beta + 3) q^{83} + ( - 2 \beta - 1) q^{85} + ( - 5 \beta + 2) q^{86} + ( - 3 \beta - 3) q^{87} - 2 \beta q^{88} + (5 \beta + 7) q^{89} - \beta q^{90} + ( - \beta - 2) q^{91} + ( - \beta - 2) q^{93} + (6 \beta + 6) q^{94} + ( - 2 \beta + 1) q^{95} + (3 \beta - 3) q^{97} + \beta q^{98} + q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 2 q^{11} + 4 q^{13} + 2 q^{15} - 8 q^{16} + 2 q^{17} - 2 q^{19} + 2 q^{21} - 2 q^{23} + 2 q^{25} + 4 q^{26} - 2 q^{27} + 6 q^{29} + 4 q^{31} - 2 q^{33} + 8 q^{34} + 2 q^{35} - 4 q^{37} + 8 q^{38} - 4 q^{39} + 8 q^{41} - 10 q^{43} - 2 q^{45} + 8 q^{46} + 12 q^{47} + 8 q^{48} + 2 q^{49} - 2 q^{51} + 6 q^{53} - 2 q^{55} + 2 q^{57} + 12 q^{58} + 26 q^{59} + 10 q^{61} + 4 q^{62} - 2 q^{63} + 16 q^{64} - 4 q^{65} - 12 q^{67} + 2 q^{69} - 8 q^{71} + 12 q^{74} - 2 q^{75} - 2 q^{77} - 4 q^{78} - 8 q^{79} + 8 q^{80} + 2 q^{81} - 20 q^{82} + 6 q^{83} - 2 q^{85} + 4 q^{86} - 6 q^{87} + 14 q^{89} - 4 q^{91} - 4 q^{93} + 12 q^{94} + 2 q^{95} - 6 q^{97} + 2 q^{99}+O(q^{100}) \) 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
−1.41421
1.41421
−1.41421 −1.00000 0 −1.00000 1.41421 −1.00000 2.82843 1.00000 1.41421
1.2 1.41421 −1.00000 0 −1.00000 −1.41421 −1.00000 −2.82843 1.00000 −1.41421
\(n\): 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.p 2
3.b odd 2 1 3465.2.a.w 2
5.b even 2 1 5775.2.a.bi 2
7.b odd 2 1 8085.2.a.bf 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1155.2.a.p 2 1.a even 1 1 trivial
3465.2.a.w 2 3.b odd 2 1
5775.2.a.bi 2 5.b even 2 1
8085.2.a.bf 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1155))\):

\( T_{2}^{2} - 2 \) Copy content Toggle raw display
\( T_{13}^{2} - 4T_{13} + 2 \) Copy content Toggle raw display
\( T_{17}^{2} - 2T_{17} - 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 2 \) Copy content Toggle raw display
$3$ \( (T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 4T + 2 \) Copy content Toggle raw display
$17$ \( T^{2} - 2T - 7 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T - 7 \) Copy content Toggle raw display
$23$ \( T^{2} + 2T - 7 \) Copy content Toggle raw display
$29$ \( T^{2} - 6T - 9 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T + 2 \) Copy content Toggle raw display
$37$ \( T^{2} + 4T - 14 \) Copy content Toggle raw display
$41$ \( T^{2} - 8T - 34 \) Copy content Toggle raw display
$43$ \( T^{2} + 10T + 23 \) Copy content Toggle raw display
$47$ \( T^{2} - 12T + 18 \) Copy content Toggle raw display
$53$ \( T^{2} - 6T - 23 \) Copy content Toggle raw display
$59$ \( T^{2} - 26T + 167 \) Copy content Toggle raw display
$61$ \( T^{2} - 10T - 7 \) Copy content Toggle raw display
$67$ \( (T + 6)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} + 8T - 34 \) Copy content Toggle raw display
$73$ \( T^{2} - 72 \) Copy content Toggle raw display
$79$ \( T^{2} + 8T - 2 \) Copy content Toggle raw display
$83$ \( T^{2} - 6T - 191 \) Copy content Toggle raw display
$89$ \( T^{2} - 14T - 1 \) Copy content Toggle raw display
$97$ \( T^{2} + 6T - 9 \) Copy content Toggle raw display
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