Properties

Label 9025.2.a.l
Level $9025$
Weight $2$
Character orbit 9025.a
Self dual yes
Analytic conductor $72.065$
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] = [9025,2,Mod(1,9025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9025, 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("9025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9025 = 5^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9025.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: \(72.0649878242\)
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: \(\Z[a_1, a_2]\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{5})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta - 1) q^{2} + (2 \beta - 1) q^{3} + 3 \beta q^{4} + ( - 3 \beta - 1) q^{6} + ( - 2 \beta - 1) q^{7} + ( - 4 \beta - 1) q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta - 1) q^{2} + (2 \beta - 1) q^{3} + 3 \beta q^{4} + ( - 3 \beta - 1) q^{6} + ( - 2 \beta - 1) q^{7} + ( - 4 \beta - 1) q^{8} + 2 q^{9} + (4 \beta - 1) q^{11} + (3 \beta + 6) q^{12} + 2 q^{13} + (5 \beta + 3) q^{14} + (3 \beta + 5) q^{16} - 3 \beta q^{17} + ( - 2 \beta - 2) q^{18} + ( - 4 \beta - 3) q^{21} + ( - 7 \beta - 3) q^{22} + ( - 2 \beta + 2) q^{23} + ( - 6 \beta - 7) q^{24} + ( - 2 \beta - 2) q^{26} + ( - 2 \beta + 1) q^{27} + ( - 9 \beta - 6) q^{28} - 6 q^{29} + (3 \beta - 3) q^{31} + ( - 3 \beta - 6) q^{32} + (2 \beta + 9) q^{33} + (6 \beta + 3) q^{34} + 6 \beta q^{36} + (3 \beta - 9) q^{37} + (4 \beta - 2) q^{39} + ( - \beta + 8) q^{41} + (11 \beta + 7) q^{42} + (9 \beta - 4) q^{43} + (9 \beta + 12) q^{44} + 2 \beta q^{46} + ( - \beta - 6) q^{47} + (13 \beta + 1) q^{48} + (8 \beta - 2) q^{49} + ( - 3 \beta - 6) q^{51} + 6 \beta q^{52} + ( - \beta + 8) q^{53} + (3 \beta + 1) q^{54} + (14 \beta + 9) q^{56} + (6 \beta + 6) q^{58} + ( - 2 \beta + 6) q^{59} + ( - 9 \beta + 9) q^{61} - 3 \beta q^{62} + ( - 4 \beta - 2) q^{63} + (6 \beta - 1) q^{64} + ( - 13 \beta - 11) q^{66} + (6 \beta - 1) q^{67} + ( - 9 \beta - 9) q^{68} + (2 \beta - 6) q^{69} + ( - 4 \beta + 11) q^{71} + ( - 8 \beta - 2) q^{72} + ( - 6 \beta - 1) q^{73} + (3 \beta + 6) q^{74} + ( - 10 \beta - 7) q^{77} + ( - 6 \beta - 2) q^{78} - q^{79} - 11 q^{81} + ( - 6 \beta - 7) q^{82} + (5 \beta - 7) q^{83} + ( - 21 \beta - 12) q^{84} + ( - 14 \beta - 5) q^{86} + ( - 12 \beta + 6) q^{87} + ( - 16 \beta - 15) q^{88} + ( - 6 \beta + 6) q^{89} + ( - 4 \beta - 2) q^{91} - 6 q^{92} + ( - 3 \beta + 9) q^{93} + (8 \beta + 7) q^{94} - 15 \beta q^{96} + (\beta - 9) q^{97} + ( - 14 \beta - 6) q^{98} + (8 \beta - 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} + 3 q^{4} - 5 q^{6} - 4 q^{7} - 6 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} + 3 q^{4} - 5 q^{6} - 4 q^{7} - 6 q^{8} + 4 q^{9} + 2 q^{11} + 15 q^{12} + 4 q^{13} + 11 q^{14} + 13 q^{16} - 3 q^{17} - 6 q^{18} - 10 q^{21} - 13 q^{22} + 2 q^{23} - 20 q^{24} - 6 q^{26} - 21 q^{28} - 12 q^{29} - 3 q^{31} - 15 q^{32} + 20 q^{33} + 12 q^{34} + 6 q^{36} - 15 q^{37} + 15 q^{41} + 25 q^{42} + q^{43} + 33 q^{44} + 2 q^{46} - 13 q^{47} + 15 q^{48} + 4 q^{49} - 15 q^{51} + 6 q^{52} + 15 q^{53} + 5 q^{54} + 32 q^{56} + 18 q^{58} + 10 q^{59} + 9 q^{61} - 3 q^{62} - 8 q^{63} + 4 q^{64} - 35 q^{66} + 4 q^{67} - 27 q^{68} - 10 q^{69} + 18 q^{71} - 12 q^{72} - 8 q^{73} + 15 q^{74} - 24 q^{77} - 10 q^{78} - 2 q^{79} - 22 q^{81} - 20 q^{82} - 9 q^{83} - 45 q^{84} - 24 q^{86} - 46 q^{88} + 6 q^{89} - 8 q^{91} - 12 q^{92} + 15 q^{93} + 22 q^{94} - 15 q^{96} - 17 q^{97} - 26 q^{98} + 4 q^{99}+O(q^{100}) \) 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
−2.61803 2.23607 4.85410 0 −5.85410 −4.23607 −7.47214 2.00000 0
1.2 −0.381966 −2.23607 −1.85410 0 0.854102 0.236068 1.47214 2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( -1 \)
\(19\) \( -1 \)

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 9025.2.a.l 2
5.b even 2 1 9025.2.a.u yes 2
19.b odd 2 1 9025.2.a.t yes 2
95.d odd 2 1 9025.2.a.m yes 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
9025.2.a.l 2 1.a even 1 1 trivial
9025.2.a.m yes 2 95.d odd 2 1
9025.2.a.t yes 2 19.b odd 2 1
9025.2.a.u yes 2 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(9025))\):

\( T_{2}^{2} + 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{2} - 5 \) Copy content Toggle raw display
\( T_{7}^{2} + 4T_{7} - 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 2T_{11} - 19 \) Copy content Toggle raw display
\( T_{29} + 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 3T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - 5 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 4T - 1 \) Copy content Toggle raw display
$11$ \( T^{2} - 2T - 19 \) Copy content Toggle raw display
$13$ \( (T - 2)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 3T - 9 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 3T - 9 \) Copy content Toggle raw display
$37$ \( T^{2} + 15T + 45 \) Copy content Toggle raw display
$41$ \( T^{2} - 15T + 55 \) Copy content Toggle raw display
$43$ \( T^{2} - T - 101 \) Copy content Toggle raw display
$47$ \( T^{2} + 13T + 41 \) Copy content Toggle raw display
$53$ \( T^{2} - 15T + 55 \) Copy content Toggle raw display
$59$ \( T^{2} - 10T + 20 \) Copy content Toggle raw display
$61$ \( T^{2} - 9T - 81 \) Copy content Toggle raw display
$67$ \( T^{2} - 4T - 41 \) Copy content Toggle raw display
$71$ \( T^{2} - 18T + 61 \) Copy content Toggle raw display
$73$ \( T^{2} + 8T - 29 \) Copy content Toggle raw display
$79$ \( (T + 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 9T - 11 \) Copy content Toggle raw display
$89$ \( T^{2} - 6T - 36 \) Copy content Toggle raw display
$97$ \( T^{2} + 17T + 71 \) Copy content Toggle raw display
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