Properties

Label 1183.2.a.i
Level $1183$
Weight $2$
Character orbit 1183.a
Self dual yes
Analytic conductor $9.446$
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] = [1183,2,Mod(1,1183)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1183, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1183.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1183.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: \(9.44630255912\)
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: no (minimal twist has level 91)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + ( - \beta_{2} + \beta_1 - 1) q^{3} + (\beta_{2} + 1) q^{4} + (\beta_1 - 1) q^{5} + (2 \beta_1 - 2) q^{6} + q^{7} + ( - \beta_{2} - 1) q^{8} + ( - 2 \beta_1 + 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + ( - \beta_{2} + \beta_1 - 1) q^{3} + (\beta_{2} + 1) q^{4} + (\beta_1 - 1) q^{5} + (2 \beta_1 - 2) q^{6} + q^{7} + ( - \beta_{2} - 1) q^{8} + ( - 2 \beta_1 + 3) q^{9} + ( - \beta_{2} + \beta_1 - 3) q^{10} + ( - \beta_{2} + \beta_1 - 1) q^{11} - 4 q^{12} - \beta_1 q^{14} + (\beta_{2} - 3 \beta_1 + 3) q^{15} + ( - \beta_{2} + 2 \beta_1 - 1) q^{16} + (\beta_{2} + \beta_1 + 1) q^{17} + (2 \beta_{2} - 3 \beta_1 + 6) q^{18} + (\beta_1 + 1) q^{19} + 2 \beta_1 q^{20} + ( - \beta_{2} + \beta_1 - 1) q^{21} + (2 \beta_1 - 2) q^{22} + ( - \beta_{2} - 2 \beta_1 + 4) q^{23} + 4 q^{24} + (\beta_{2} - 2 \beta_1 - 1) q^{25} + (4 \beta_1 - 4) q^{27} + (\beta_{2} + 1) q^{28} + (\beta_{2} + 8) q^{29} + (2 \beta_{2} - 4 \beta_1 + 8) q^{30} + ( - 2 \beta_{2} + \beta_1 + 1) q^{31} + (\beta_{2} + 2 \beta_1 - 3) q^{32} + ( - 2 \beta_1 + 6) q^{33} + ( - 2 \beta_{2} - 2 \beta_1 - 4) q^{34} + (\beta_1 - 1) q^{35} + (\beta_{2} - 4 \beta_1 + 1) q^{36} + ( - \beta_{2} - 3 \beta_1 + 1) q^{37} + ( - \beta_{2} - \beta_1 - 3) q^{38} - 2 \beta_1 q^{40} + (2 \beta_{2} - 2 \beta_1) q^{41} + (2 \beta_1 - 2) q^{42} + ( - 3 \beta_{2} - 2 \beta_1 + 4) q^{43} - 4 q^{44} + ( - 2 \beta_{2} + 5 \beta_1 - 9) q^{45} + (3 \beta_{2} - 3 \beta_1 + 7) q^{46} + (4 \beta_{2} - \beta_1 + 3) q^{47} + ( - 4 \beta_1 + 8) q^{48} + q^{49} + (\beta_{2} + 5) q^{50} + ( - 2 \beta_1 - 2) q^{51} + ( - 3 \beta_{2} + 2 \beta_1 + 2) q^{53} + ( - 4 \beta_{2} + 4 \beta_1 - 12) q^{54} + (\beta_{2} - 3 \beta_1 + 3) q^{55} + ( - \beta_{2} - 1) q^{56} + ( - \beta_{2} - \beta_1 + 1) q^{57} + ( - \beta_{2} - 9 \beta_1 - 1) q^{58} + ( - 4 \beta_{2} - 2 \beta_1 + 2) q^{59} + ( - 4 \beta_1 + 4) q^{60} - 2 q^{61} + (\beta_{2} + \beta_1 - 1) q^{62} + ( - 2 \beta_1 + 3) q^{63} + ( - \beta_{2} - 2 \beta_1 - 5) q^{64} + (2 \beta_{2} - 6 \beta_1 + 6) q^{66} + ( - 4 \beta_{2} + 6 \beta_1 + 2) q^{67} + (2 \beta_{2} + 4 \beta_1 + 6) q^{68} + ( - 5 \beta_{2} + 9 \beta_1 - 5) q^{69} + ( - \beta_{2} + \beta_1 - 3) q^{70} + (\beta_{2} - 3 \beta_1 + 3) q^{71} + ( - \beta_{2} + 4 \beta_1 - 1) q^{72} + (4 \beta_{2} + \beta_1 + 3) q^{73} + (4 \beta_{2} + 10) q^{74} + (2 \beta_{2} + 2 \beta_1 - 6) q^{75} + (2 \beta_{2} + 2 \beta_1 + 2) q^{76} + ( - \beta_{2} + \beta_1 - 1) q^{77} + ( - \beta_{2} + 4 \beta_1 - 6) q^{79} + (2 \beta_{2} - 4 \beta_1 + 6) q^{80} + (4 \beta_{2} - 6 \beta_1 + 3) q^{81} + ( - 2 \beta_1 + 4) q^{82} + ( - 4 \beta_{2} + 9 \beta_1 + 1) q^{83} - 4 q^{84} + (\beta_{2} + \beta_1 + 3) q^{85} + (5 \beta_{2} - \beta_1 + 9) q^{86} + ( - 7 \beta_{2} + 7 \beta_1 - 11) q^{87} + 4 q^{88} + (2 \beta_{2} - 5 \beta_1 + 1) q^{89} + ( - 3 \beta_{2} + 11 \beta_1 - 13) q^{90} + (2 \beta_{2} - 6 \beta_1 - 2) q^{92} + ( - 3 \beta_{2} + \beta_1 + 7) q^{93} + ( - 3 \beta_{2} - 7 \beta_1 - 1) q^{94} + (\beta_{2} + 2) q^{95} + (4 \beta_{2} - 8 \beta_1 + 4) q^{96} + (\beta_1 + 3) q^{97} - \beta_1 q^{98} + ( - 3 \beta_{2} + 7 \beta_1 - 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} - 2 q^{3} + 3 q^{4} - 2 q^{5} - 4 q^{6} + 3 q^{7} - 3 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} - 2 q^{3} + 3 q^{4} - 2 q^{5} - 4 q^{6} + 3 q^{7} - 3 q^{8} + 7 q^{9} - 8 q^{10} - 2 q^{11} - 12 q^{12} - q^{14} + 6 q^{15} - q^{16} + 4 q^{17} + 15 q^{18} + 4 q^{19} + 2 q^{20} - 2 q^{21} - 4 q^{22} + 10 q^{23} + 12 q^{24} - 5 q^{25} - 8 q^{27} + 3 q^{28} + 24 q^{29} + 20 q^{30} + 4 q^{31} - 7 q^{32} + 16 q^{33} - 14 q^{34} - 2 q^{35} - q^{36} - 10 q^{38} - 2 q^{40} - 2 q^{41} - 4 q^{42} + 10 q^{43} - 12 q^{44} - 22 q^{45} + 18 q^{46} + 8 q^{47} + 20 q^{48} + 3 q^{49} + 15 q^{50} - 8 q^{51} + 8 q^{53} - 32 q^{54} + 6 q^{55} - 3 q^{56} + 2 q^{57} - 12 q^{58} + 4 q^{59} + 8 q^{60} - 6 q^{61} - 2 q^{62} + 7 q^{63} - 17 q^{64} + 12 q^{66} + 12 q^{67} + 22 q^{68} - 6 q^{69} - 8 q^{70} + 6 q^{71} + q^{72} + 10 q^{73} + 30 q^{74} - 16 q^{75} + 8 q^{76} - 2 q^{77} - 14 q^{79} + 14 q^{80} + 3 q^{81} + 10 q^{82} + 12 q^{83} - 12 q^{84} + 10 q^{85} + 26 q^{86} - 26 q^{87} + 12 q^{88} - 2 q^{89} - 28 q^{90} - 12 q^{92} + 22 q^{93} - 10 q^{94} + 6 q^{95} + 4 q^{96} + 10 q^{97} - q^{98} - 14 q^{99}+O(q^{100}) \) 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
2.34292
0.470683
−1.81361
−2.34292 −1.14637 3.48929 1.34292 2.68585 1.00000 −3.48929 −1.68585 −3.14637
1.2 −0.470683 2.24914 −1.77846 −0.529317 −1.05863 1.00000 1.77846 2.05863 0.249141
1.3 1.81361 −3.10278 1.28917 −2.81361 −5.62721 1.00000 −1.28917 6.62721 −5.10278
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(7\) \( -1 \)
\(13\) \( +1 \)

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 1183.2.a.i 3
7.b odd 2 1 8281.2.a.bg 3
13.b even 2 1 91.2.a.d 3
13.d odd 4 2 1183.2.c.f 6
39.d odd 2 1 819.2.a.i 3
52.b odd 2 1 1456.2.a.t 3
65.d even 2 1 2275.2.a.m 3
91.b odd 2 1 637.2.a.j 3
91.r even 6 2 637.2.e.j 6
91.s odd 6 2 637.2.e.i 6
104.e even 2 1 5824.2.a.by 3
104.h odd 2 1 5824.2.a.bs 3
273.g even 2 1 5733.2.a.x 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.d 3 13.b even 2 1
637.2.a.j 3 91.b odd 2 1
637.2.e.i 6 91.s odd 6 2
637.2.e.j 6 91.r even 6 2
819.2.a.i 3 39.d odd 2 1
1183.2.a.i 3 1.a even 1 1 trivial
1183.2.c.f 6 13.d odd 4 2
1456.2.a.t 3 52.b odd 2 1
2275.2.a.m 3 65.d even 2 1
5733.2.a.x 3 273.g even 2 1
5824.2.a.bs 3 104.h odd 2 1
5824.2.a.by 3 104.e even 2 1
8281.2.a.bg 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1183))\):

\( T_{2}^{3} + T_{2}^{2} - 4T_{2} - 2 \) Copy content Toggle raw display
\( T_{11}^{3} + 2T_{11}^{2} - 6T_{11} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} + T^{2} - 4T - 2 \) Copy content Toggle raw display
$3$ \( T^{3} + 2 T^{2} + \cdots - 8 \) Copy content Toggle raw display
$5$ \( T^{3} + 2 T^{2} + \cdots - 2 \) Copy content Toggle raw display
$7$ \( (T - 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 2 T^{2} + \cdots - 8 \) Copy content Toggle raw display
$13$ \( T^{3} \) Copy content Toggle raw display
$17$ \( T^{3} - 4 T^{2} + \cdots - 4 \) Copy content Toggle raw display
$19$ \( T^{3} - 4T^{2} + T + 4 \) Copy content Toggle raw display
$23$ \( T^{3} - 10 T^{2} + \cdots + 136 \) Copy content Toggle raw display
$29$ \( T^{3} - 24 T^{2} + \cdots - 454 \) Copy content Toggle raw display
$31$ \( T^{3} - 4 T^{2} + \cdots - 16 \) Copy content Toggle raw display
$37$ \( T^{3} - 58T + 124 \) Copy content Toggle raw display
$41$ \( T^{3} + 2 T^{2} + \cdots + 8 \) Copy content Toggle raw display
$43$ \( T^{3} - 10 T^{2} + \cdots + 628 \) Copy content Toggle raw display
$47$ \( T^{3} - 8 T^{2} + \cdots + 544 \) Copy content Toggle raw display
$53$ \( T^{3} - 8 T^{2} + \cdots - 22 \) Copy content Toggle raw display
$59$ \( T^{3} - 4 T^{2} + \cdots + 688 \) Copy content Toggle raw display
$61$ \( (T + 2)^{3} \) Copy content Toggle raw display
$67$ \( T^{3} - 12 T^{2} + \cdots + 976 \) Copy content Toggle raw display
$71$ \( T^{3} - 6 T^{2} + \cdots - 16 \) Copy content Toggle raw display
$73$ \( T^{3} - 10 T^{2} + \cdots + 274 \) Copy content Toggle raw display
$79$ \( T^{3} + 14 T^{2} + \cdots - 16 \) Copy content Toggle raw display
$83$ \( T^{3} - 12 T^{2} + \cdots + 3268 \) Copy content Toggle raw display
$89$ \( T^{3} + 2 T^{2} + \cdots - 422 \) Copy content Toggle raw display
$97$ \( T^{3} - 10 T^{2} + \cdots - 22 \) Copy content Toggle raw display
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