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Properties

Label 1161.2.a.m
Level $1161$
Weight $2$
Character orbit 1161.a
Self dual yes
Analytic conductor $9.271$
Analytic rank $0$
Dimension $10$
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] = [1161,2,Mod(1,1161)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1161, 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("1161.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1161 = 3^{3} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1161.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: \(9.27063167467\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 19x^{8} + 125x^{6} - 337x^{4} + 346x^{2} - 108 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + \beta_{7} q^{5} + ( - \beta_{8} - \beta_{3} + 1) q^{7} + ( - \beta_{7} + \beta_{6} + 2 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{2} + 2) q^{4} + \beta_{7} q^{5} + ( - \beta_{8} - \beta_{3} + 1) q^{7} + ( - \beta_{7} + \beta_{6} + 2 \beta_1) q^{8} + ( - \beta_{5} + \beta_{3} - \beta_{2}) q^{10} + (\beta_{9} + \beta_{4}) q^{11} + ( - \beta_{8} - \beta_{2}) q^{13} + ( - \beta_{7} + \beta_{4} + 2 \beta_1) q^{14} + (\beta_{8} + \beta_{5} + 3 \beta_{2} + 4) q^{16} - \beta_{4} q^{17} + (\beta_{3} + 2) q^{19} + ( - 2 \beta_{9} + 4 \beta_{7} + \cdots - \beta_1) q^{20}+ \cdots + (3 \beta_{9} - 5 \beta_{7} + \cdots + \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 18 q^{4} + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 18 q^{4} + 12 q^{7} + 34 q^{16} + 16 q^{19} + 20 q^{22} + 22 q^{25} + 60 q^{28} + 20 q^{31} - 8 q^{34} + 12 q^{37} - 64 q^{40} + 10 q^{43} + 22 q^{46} + 34 q^{49} - 52 q^{52} + 26 q^{55} - 12 q^{58} + 28 q^{61} + 102 q^{64} - 4 q^{67} - 80 q^{70} + 12 q^{73} + 16 q^{76} + 16 q^{79} + 28 q^{82} + 8 q^{85} - 4 q^{88} + 2 q^{91} - 44 q^{94} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 19x^{8} + 125x^{6} - 337x^{4} + 346x^{2} - 108 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{8} + 14\nu^{6} - 55\nu^{4} + 50\nu^{2} ) / 12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 14\nu^{5} + 55\nu^{3} - 50\nu ) / 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{8} + 17\nu^{6} - 88\nu^{4} + 134\nu^{2} - 30 ) / 6 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{9} - 18\nu^{7} + 105\nu^{5} - 204\nu^{3} + 44\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{9} - 18\nu^{7} + 105\nu^{5} - 216\nu^{3} + 116\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{8} - 17\nu^{6} + 94\nu^{4} - 188\nu^{2} + 102 ) / 6 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{9} - 18\nu^{7} + 108\nu^{5} - 243\nu^{3} + 158\nu ) / 6 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} + \beta_{6} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} + \beta_{5} + 9\beta_{2} + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{9} - 13\beta_{7} + 9\beta_{6} + 40\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 11\beta_{8} + 13\beta_{5} - 4\beta_{3} + 71\beta_{2} + 162 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 28\beta_{9} - 127\beta_{7} + 71\beta_{6} + 6\beta_{4} + 280\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 99\beta_{8} + 127\beta_{5} - 68\beta_{3} + 549\beta_{2} + 1148 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 294\beta_{9} - 1125\beta_{7} + 549\beta_{6} + 108\beta_{4} + 2020\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
−2.81532
−2.43071
−1.84006
−1.10575
−0.746379
0.746379
1.10575
1.84006
2.43071
2.81532
−2.81532 0 5.92605 3.71505 0 4.04516 −11.0531 0 −10.4591
1.2 −2.43071 0 3.90834 −2.28282 0 −0.190638 −4.63863 0 5.54886
1.3 −1.84006 0 1.38583 −3.23863 0 2.66351 1.13011 0 5.95928
1.4 −1.10575 0 −0.777307 2.00844 0 3.50740 3.07102 0 −2.22084
1.5 −0.746379 0 −1.44292 −1.56991 0 −4.02544 2.56972 0 1.17175
1.6 0.746379 0 −1.44292 1.56991 0 −4.02544 −2.56972 0 1.17175
1.7 1.10575 0 −0.777307 −2.00844 0 3.50740 −3.07102 0 −2.22084
1.8 1.84006 0 1.38583 3.23863 0 2.66351 −1.13011 0 5.95928
1.9 2.43071 0 3.90834 2.28282 0 −0.190638 4.63863 0 5.54886
1.10 2.81532 0 5.92605 −3.71505 0 4.04516 11.0531 0 −10.4591
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(43\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1161.2.a.m ✓ 10
3.b odd 2 1 inner 1161.2.a.m ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1161.2.a.m ✓ 10 1.a even 1 1 trivial
1161.2.a.m ✓ 10 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1161))\):

\( T_{2}^{10} - 19T_{2}^{8} + 125T_{2}^{6} - 337T_{2}^{4} + 346T_{2}^{2} - 108 \) Copy content Toggle raw display
\( T_{5}^{10} - 36T_{5}^{8} + 473T_{5}^{6} - 2811T_{5}^{4} + 7600T_{5}^{2} - 7500 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 19 T^{8} + \cdots - 108 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} - 36 T^{8} + \cdots - 7500 \) Copy content Toggle raw display
$7$ \( (T^{5} - 6 T^{4} - 8 T^{3} + \cdots - 29)^{2} \) Copy content Toggle raw display
$11$ \( T^{10} - 76 T^{8} + \cdots - 139968 \) Copy content Toggle raw display
$13$ \( (T^{5} - 24 T^{3} + \cdots + 11)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} - 42 T^{8} + \cdots - 768 \) Copy content Toggle raw display
$19$ \( (T^{5} - 8 T^{4} + 6 T^{3} + \cdots - 71)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} - 171 T^{8} + \cdots - 817452 \) Copy content Toggle raw display
$29$ \( T^{10} - 156 T^{8} + \cdots - 6377292 \) Copy content Toggle raw display
$31$ \( (T^{5} - 10 T^{4} + \cdots - 4896)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} - 6 T^{4} - 5 T^{3} + \cdots - 72)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} - 268 T^{8} + \cdots - 559872 \) Copy content Toggle raw display
$43$ \( (T - 1)^{10} \) Copy content Toggle raw display
$47$ \( T^{10} - 126 T^{8} + \cdots - 2239488 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots - 128471808 \) Copy content Toggle raw display
$59$ \( T^{10} - 333 T^{8} + \cdots - 82309932 \) Copy content Toggle raw display
$61$ \( (T^{5} - 14 T^{4} + 59 T^{3} + \cdots + 8)^{2} \) Copy content Toggle raw display
$67$ \( (T^{5} + 2 T^{4} + \cdots - 576)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} - 374 T^{8} + \cdots - 18930432 \) Copy content Toggle raw display
$73$ \( (T^{5} - 6 T^{4} - 5 T^{3} + \cdots - 72)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} - 8 T^{4} + \cdots - 1000)^{2} \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 2066872512 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots - 1077762348 \) Copy content Toggle raw display
$97$ \( (T^{5} + 8 T^{4} - 90 T^{3} + \cdots + 67)^{2} \) Copy content Toggle raw display
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