Properties

Label 1150.2.a.q
Level $1150$
Weight $2$
Character orbit 1150.a
Self dual yes
Analytic conductor $9.183$
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] = [1150,2,Mod(1,1150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1150, 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("1150.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1150 = 2 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1150.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: \(9.18279623245\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1101.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 9x + 12 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 230)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 9x + 12 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + \nu - 7 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - \beta _1 + 7 \) 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
2.68740
1.43163
−3.11903
−1.00000 −2.68740 1.00000 0 2.68740 4.59692 −1.00000 4.22212 0
1.2 −1.00000 −1.43163 1.00000 0 1.43163 −3.08719 −1.00000 −0.950444 0
1.3 −1.00000 3.11903 1.00000 0 −3.11903 −4.50973 −1.00000 6.72833 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( +1 \)
\(23\) \( -1 \)

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 1150.2.a.q 3
4.b odd 2 1 9200.2.a.cf 3
5.b even 2 1 230.2.a.d 3
5.c odd 4 2 1150.2.b.j 6
15.d odd 2 1 2070.2.a.z 3
20.d odd 2 1 1840.2.a.r 3
40.e odd 2 1 7360.2.a.ce 3
40.f even 2 1 7360.2.a.bz 3
115.c odd 2 1 5290.2.a.r 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.a.d 3 5.b even 2 1
1150.2.a.q 3 1.a even 1 1 trivial
1150.2.b.j 6 5.c odd 4 2
1840.2.a.r 3 20.d odd 2 1
2070.2.a.z 3 15.d odd 2 1
5290.2.a.r 3 115.c odd 2 1
7360.2.a.bz 3 40.f even 2 1
7360.2.a.ce 3 40.e odd 2 1
9200.2.a.cf 3 4.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1150))\):

\( T_{3}^{3} + T_{3}^{2} - 9T_{3} - 12 \) Copy content Toggle raw display
\( T_{7}^{3} + 3T_{7}^{2} - 21T_{7} - 64 \) Copy content Toggle raw display
\( T_{11}^{3} - 3T_{11}^{2} - 39T_{11} + 144 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{3} \) Copy content Toggle raw display
$3$ \( T^{3} + T^{2} - 9T - 12 \) Copy content Toggle raw display
$5$ \( T^{3} \) Copy content Toggle raw display
$7$ \( T^{3} + 3 T^{2} + \cdots - 64 \) Copy content Toggle raw display
$11$ \( T^{3} - 3 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$13$ \( T^{3} - T^{2} + \cdots + 18 \) Copy content Toggle raw display
$17$ \( T^{3} - 7 T^{2} + \cdots + 18 \) Copy content Toggle raw display
$19$ \( T^{3} - 3 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$23$ \( (T - 1)^{3} \) Copy content Toggle raw display
$29$ \( T^{3} + 4 T^{2} + \cdots + 24 \) Copy content Toggle raw display
$31$ \( T^{3} + 5 T^{2} + \cdots - 8 \) Copy content Toggle raw display
$37$ \( T^{3} - 2 T^{2} + \cdots + 32 \) Copy content Toggle raw display
$41$ \( T^{3} - T^{2} + \cdots + 186 \) Copy content Toggle raw display
$43$ \( (T + 8)^{3} \) Copy content Toggle raw display
$47$ \( T^{3} - 14 T^{2} + \cdots + 288 \) Copy content Toggle raw display
$53$ \( (T - 6)^{3} \) Copy content Toggle raw display
$59$ \( T^{3} - 14 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$61$ \( T^{3} - T^{2} + \cdots + 526 \) Copy content Toggle raw display
$67$ \( T^{3} + 8 T^{2} + \cdots - 384 \) Copy content Toggle raw display
$71$ \( T^{3} - 11 T^{2} + \cdots - 24 \) Copy content Toggle raw display
$73$ \( T^{3} - 8 T^{2} + \cdots + 248 \) Copy content Toggle raw display
$79$ \( T^{3} + 4 T^{2} + \cdots - 1152 \) Copy content Toggle raw display
$83$ \( T^{3} + 8 T^{2} + \cdots - 96 \) Copy content Toggle raw display
$89$ \( T^{3} - 18 T^{2} + \cdots + 1152 \) Copy content Toggle raw display
$97$ \( T^{3} - 33 T^{2} + \cdots - 166 \) Copy content Toggle raw display
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