Properties

Label 9025.2.a.s
Level $9025$
Weight $2$
Character orbit 9025.a
Self dual yes
Analytic conductor $72.065$
Analytic rank $0$
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: \(0\)
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: no (minimal twist has level 361)
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.

\(f(q)\) \(=\) \( q + \beta q^{2} + ( - \beta + 2) q^{3} + (\beta - 1) q^{4} + (\beta - 1) q^{6} - 3 q^{7} + ( - 2 \beta + 1) q^{8} + ( - 3 \beta + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta q^{2} + ( - \beta + 2) q^{3} + (\beta - 1) q^{4} + (\beta - 1) q^{6} - 3 q^{7} + ( - 2 \beta + 1) q^{8} + ( - 3 \beta + 2) q^{9} - \beta q^{11} + (2 \beta - 3) q^{12} - q^{13} - 3 \beta q^{14} - 3 \beta q^{16} + (2 \beta - 4) q^{17} + ( - \beta - 3) q^{18} + (3 \beta - 6) q^{21} + ( - \beta - 1) q^{22} + (\beta - 7) q^{23} + ( - 3 \beta + 4) q^{24} - \beta q^{26} + ( - 2 \beta + 1) q^{27} + ( - 3 \beta + 3) q^{28} + (\beta + 2) q^{29} + (3 \beta + 4) q^{31} + (\beta - 5) q^{32} + ( - \beta + 1) q^{33} + ( - 2 \beta + 2) q^{34} + (2 \beta - 5) q^{36} + (3 \beta + 4) q^{37} + (\beta - 2) q^{39} + 3 q^{41} + ( - 3 \beta + 3) q^{42} + ( - 3 \beta + 5) q^{43} - q^{44} + ( - 6 \beta + 1) q^{46} - 3 q^{47} + ( - 3 \beta + 3) q^{48} + 2 q^{49} + (6 \beta - 10) q^{51} + ( - \beta + 1) q^{52} + ( - 7 \beta + 5) q^{53} + ( - \beta - 2) q^{54} + (6 \beta - 3) q^{56} + (3 \beta + 1) q^{58} + ( - 7 \beta + 11) q^{59} + ( - 2 \beta - 7) q^{61} + (7 \beta + 3) q^{62} + (9 \beta - 6) q^{63} + (2 \beta + 1) q^{64} - q^{66} - 7 q^{67} + ( - 4 \beta + 6) q^{68} + (8 \beta - 15) q^{69} + (4 \beta + 1) q^{71} + ( - \beta + 8) q^{72} + (6 \beta - 7) q^{73} + (7 \beta + 3) q^{74} + 3 \beta q^{77} + ( - \beta + 1) q^{78} + ( - 12 \beta + 6) q^{79} + (6 \beta - 2) q^{81} + 3 \beta q^{82} + ( - 4 \beta - 2) q^{83} + ( - 6 \beta + 9) q^{84} + (2 \beta - 3) q^{86} + ( - \beta + 3) q^{87} + (\beta + 2) q^{88} + ( - 2 \beta + 11) q^{89} + 3 q^{91} + ( - 7 \beta + 8) q^{92} + ( - \beta + 5) q^{93} - 3 \beta q^{94} + (6 \beta - 11) q^{96} + (3 \beta + 9) q^{97} + 2 \beta q^{98} + (\beta + 3) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 3 q^{3} - q^{4} - q^{6} - 6 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 3 q^{3} - q^{4} - q^{6} - 6 q^{7} + q^{9} - q^{11} - 4 q^{12} - 2 q^{13} - 3 q^{14} - 3 q^{16} - 6 q^{17} - 7 q^{18} - 9 q^{21} - 3 q^{22} - 13 q^{23} + 5 q^{24} - q^{26} + 3 q^{28} + 5 q^{29} + 11 q^{31} - 9 q^{32} + q^{33} + 2 q^{34} - 8 q^{36} + 11 q^{37} - 3 q^{39} + 6 q^{41} + 3 q^{42} + 7 q^{43} - 2 q^{44} - 4 q^{46} - 6 q^{47} + 3 q^{48} + 4 q^{49} - 14 q^{51} + q^{52} + 3 q^{53} - 5 q^{54} + 5 q^{58} + 15 q^{59} - 16 q^{61} + 13 q^{62} - 3 q^{63} + 4 q^{64} - 2 q^{66} - 14 q^{67} + 8 q^{68} - 22 q^{69} + 6 q^{71} + 15 q^{72} - 8 q^{73} + 13 q^{74} + 3 q^{77} + q^{78} + 2 q^{81} + 3 q^{82} - 8 q^{83} + 12 q^{84} - 4 q^{86} + 5 q^{87} + 5 q^{88} + 20 q^{89} + 6 q^{91} + 9 q^{92} + 9 q^{93} - 3 q^{94} - 16 q^{96} + 21 q^{97} + 2 q^{98} + 7 q^{99}+O(q^{100}) \) 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
−0.618034
1.61803
−0.618034 2.61803 −1.61803 0 −1.61803 −3.00000 2.23607 3.85410 0
1.2 1.61803 0.381966 0.618034 0 0.618034 −3.00000 −2.23607 −2.85410 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.s 2
5.b even 2 1 361.2.a.c 2
15.d odd 2 1 3249.2.a.o 2
19.b odd 2 1 9025.2.a.n 2
20.d odd 2 1 5776.2.a.bg 2
95.d odd 2 1 361.2.a.f yes 2
95.h odd 6 2 361.2.c.d 4
95.i even 6 2 361.2.c.g 4
95.o odd 18 6 361.2.e.j 12
95.p even 18 6 361.2.e.i 12
285.b even 2 1 3249.2.a.i 2
380.d even 2 1 5776.2.a.s 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
361.2.a.c 2 5.b even 2 1
361.2.a.f yes 2 95.d odd 2 1
361.2.c.d 4 95.h odd 6 2
361.2.c.g 4 95.i even 6 2
361.2.e.i 12 95.p even 18 6
361.2.e.j 12 95.o odd 18 6
3249.2.a.i 2 285.b even 2 1
3249.2.a.o 2 15.d odd 2 1
5776.2.a.s 2 380.d even 2 1
5776.2.a.bg 2 20.d odd 2 1
9025.2.a.n 2 19.b odd 2 1
9025.2.a.s 2 1.a even 1 1 trivial

Hecke kernels

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

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$3$ \( T^{2} - 3T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 3)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + T - 1 \) Copy content Toggle raw display
$13$ \( (T + 1)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 6T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 13T + 41 \) Copy content Toggle raw display
$29$ \( T^{2} - 5T + 5 \) Copy content Toggle raw display
$31$ \( T^{2} - 11T + 19 \) Copy content Toggle raw display
$37$ \( T^{2} - 11T + 19 \) Copy content Toggle raw display
$41$ \( (T - 3)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 7T + 1 \) Copy content Toggle raw display
$47$ \( (T + 3)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} - 3T - 59 \) Copy content Toggle raw display
$59$ \( T^{2} - 15T - 5 \) Copy content Toggle raw display
$61$ \( T^{2} + 16T + 59 \) Copy content Toggle raw display
$67$ \( (T + 7)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} - 6T - 11 \) Copy content Toggle raw display
$73$ \( T^{2} + 8T - 29 \) Copy content Toggle raw display
$79$ \( T^{2} - 180 \) Copy content Toggle raw display
$83$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$89$ \( T^{2} - 20T + 95 \) Copy content Toggle raw display
$97$ \( T^{2} - 21T + 99 \) Copy content Toggle raw display
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