Properties

Label 1150.2.a.n
Level $1150$
Weight $2$
Character orbit 1150.a
Self dual yes
Analytic conductor $9.183$
Analytic rank $1$
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] = [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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{5}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) 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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} - \beta q^{3} + q^{4} - \beta q^{6} + (3 \beta - 3) q^{7} + q^{8} + (\beta - 2) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} - \beta q^{3} + q^{4} - \beta q^{6} + (3 \beta - 3) q^{7} + q^{8} + (\beta - 2) q^{9} + ( - \beta - 4) q^{11} - \beta q^{12} + ( - \beta - 1) q^{13} + (3 \beta - 3) q^{14} + q^{16} + ( - 3 \beta + 4) q^{17} + (\beta - 2) q^{18} + (3 \beta - 5) q^{19} - 3 q^{21} + ( - \beta - 4) q^{22} - q^{23} - \beta q^{24} + ( - \beta - 1) q^{26} + (4 \beta - 1) q^{27} + (3 \beta - 3) q^{28} - 6 \beta q^{29} + (3 \beta - 7) q^{31} + q^{32} + (5 \beta + 1) q^{33} + ( - 3 \beta + 4) q^{34} + (\beta - 2) q^{36} + 6 \beta q^{37} + (3 \beta - 5) q^{38} + (2 \beta + 1) q^{39} + ( - \beta - 4) q^{41} - 3 q^{42} + ( - 2 \beta - 8) q^{43} + ( - \beta - 4) q^{44} - q^{46} + ( - 6 \beta + 8) q^{47} - \beta q^{48} + ( - 9 \beta + 11) q^{49} + ( - \beta + 3) q^{51} + ( - \beta - 1) q^{52} + 2 q^{53} + (4 \beta - 1) q^{54} + (3 \beta - 3) q^{56} + (2 \beta - 3) q^{57} - 6 \beta q^{58} - 6 q^{59} + (3 \beta - 2) q^{61} + (3 \beta - 7) q^{62} + ( - 6 \beta + 9) q^{63} + q^{64} + (5 \beta + 1) q^{66} + ( - 2 \beta - 2) q^{67} + ( - 3 \beta + 4) q^{68} + \beta q^{69} + ( - \beta + 2) q^{71} + (\beta - 2) q^{72} + (4 \beta + 10) q^{73} + 6 \beta q^{74} + (3 \beta - 5) q^{76} + ( - 12 \beta + 9) q^{77} + (2 \beta + 1) q^{78} + ( - 6 \beta + 2) q^{79} + ( - 6 \beta + 2) q^{81} + ( - \beta - 4) q^{82} + ( - 6 \beta + 2) q^{83} - 3 q^{84} + ( - 2 \beta - 8) q^{86} + (6 \beta + 6) q^{87} + ( - \beta - 4) q^{88} + (6 \beta - 6) q^{89} - 3 \beta q^{91} - q^{92} + (4 \beta - 3) q^{93} + ( - 6 \beta + 8) q^{94} - \beta q^{96} + ( - 13 \beta + 8) q^{97} + ( - 9 \beta + 11) q^{98} + ( - 3 \beta + 7) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - q^{6} - 3 q^{7} + 2 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - q^{3} + 2 q^{4} - q^{6} - 3 q^{7} + 2 q^{8} - 3 q^{9} - 9 q^{11} - q^{12} - 3 q^{13} - 3 q^{14} + 2 q^{16} + 5 q^{17} - 3 q^{18} - 7 q^{19} - 6 q^{21} - 9 q^{22} - 2 q^{23} - q^{24} - 3 q^{26} + 2 q^{27} - 3 q^{28} - 6 q^{29} - 11 q^{31} + 2 q^{32} + 7 q^{33} + 5 q^{34} - 3 q^{36} + 6 q^{37} - 7 q^{38} + 4 q^{39} - 9 q^{41} - 6 q^{42} - 18 q^{43} - 9 q^{44} - 2 q^{46} + 10 q^{47} - q^{48} + 13 q^{49} + 5 q^{51} - 3 q^{52} + 4 q^{53} + 2 q^{54} - 3 q^{56} - 4 q^{57} - 6 q^{58} - 12 q^{59} - q^{61} - 11 q^{62} + 12 q^{63} + 2 q^{64} + 7 q^{66} - 6 q^{67} + 5 q^{68} + q^{69} + 3 q^{71} - 3 q^{72} + 24 q^{73} + 6 q^{74} - 7 q^{76} + 6 q^{77} + 4 q^{78} - 2 q^{79} - 2 q^{81} - 9 q^{82} - 2 q^{83} - 6 q^{84} - 18 q^{86} + 18 q^{87} - 9 q^{88} - 6 q^{89} - 3 q^{91} - 2 q^{92} - 2 q^{93} + 10 q^{94} - q^{96} + 3 q^{97} + 13 q^{98} + 11 q^{99}+O(q^{100}) \) 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
1.61803
−0.618034
1.00000 −1.61803 1.00000 0 −1.61803 1.85410 1.00000 −0.381966 0
1.2 1.00000 0.618034 1.00000 0 0.618034 −4.85410 1.00000 −2.61803 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.n 2
4.b odd 2 1 9200.2.a.by 2
5.b even 2 1 1150.2.a.l 2
5.c odd 4 2 230.2.b.a 4
15.e even 4 2 2070.2.d.c 4
20.d odd 2 1 9200.2.a.bo 2
20.e even 4 2 1840.2.e.c 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.2.b.a 4 5.c odd 4 2
1150.2.a.l 2 5.b even 2 1
1150.2.a.n 2 1.a even 1 1 trivial
1840.2.e.c 4 20.e even 4 2
2070.2.d.c 4 15.e even 4 2
9200.2.a.bo 2 20.d odd 2 1
9200.2.a.by 2 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(1150))\):

\( T_{3}^{2} + T_{3} - 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 3T_{7} - 9 \) Copy content Toggle raw display
\( T_{11}^{2} + 9T_{11} + 19 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 3T - 9 \) Copy content Toggle raw display
$11$ \( T^{2} + 9T + 19 \) Copy content Toggle raw display
$13$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$17$ \( T^{2} - 5T - 5 \) Copy content Toggle raw display
$19$ \( T^{2} + 7T + 1 \) Copy content Toggle raw display
$23$ \( (T + 1)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 6T - 36 \) Copy content Toggle raw display
$31$ \( T^{2} + 11T + 19 \) Copy content Toggle raw display
$37$ \( T^{2} - 6T - 36 \) Copy content Toggle raw display
$41$ \( T^{2} + 9T + 19 \) Copy content Toggle raw display
$43$ \( T^{2} + 18T + 76 \) Copy content Toggle raw display
$47$ \( T^{2} - 10T - 20 \) Copy content Toggle raw display
$53$ \( (T - 2)^{2} \) Copy content Toggle raw display
$59$ \( (T + 6)^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + T - 11 \) Copy content Toggle raw display
$67$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$71$ \( T^{2} - 3T + 1 \) Copy content Toggle raw display
$73$ \( T^{2} - 24T + 124 \) Copy content Toggle raw display
$79$ \( T^{2} + 2T - 44 \) Copy content Toggle raw display
$83$ \( T^{2} + 2T - 44 \) Copy content Toggle raw display
$89$ \( T^{2} + 6T - 36 \) Copy content Toggle raw display
$97$ \( T^{2} - 3T - 209 \) Copy content Toggle raw display
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