Properties

Label 9025.2.a.r
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: \(\Z[a_1, \ldots, a_{13}]\)
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.

\(f(q)\) \(=\) \( q + ( - 2 \beta + 1) q^{2} + 2 q^{3} + 3 q^{4} + ( - 4 \beta + 2) q^{6} + ( - 2 \beta + 2) q^{7} + ( - 2 \beta + 1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 2 \beta + 1) q^{2} + 2 q^{3} + 3 q^{4} + ( - 4 \beta + 2) q^{6} + ( - 2 \beta + 2) q^{7} + ( - 2 \beta + 1) q^{8} + q^{9} + (2 \beta + 2) q^{11} + 6 q^{12} + ( - 3 \beta + 3) q^{13} + ( - 2 \beta + 6) q^{14} - q^{16} + ( - \beta + 1) q^{17} + ( - 2 \beta + 1) q^{18} + ( - 4 \beta + 4) q^{21} + ( - 6 \beta - 2) q^{22} + (4 \beta - 2) q^{23} + ( - 4 \beta + 2) q^{24} + ( - 3 \beta + 9) q^{26} - 4 q^{27} + ( - 6 \beta + 6) q^{28} + ( - 5 \beta + 1) q^{29} + 6 q^{31} + (6 \beta - 3) q^{32} + (4 \beta + 4) q^{33} + ( - \beta + 3) q^{34} + 3 q^{36} + (3 \beta + 4) q^{37} + ( - 6 \beta + 6) q^{39} + ( - 5 \beta + 4) q^{41} + ( - 4 \beta + 12) q^{42} + (4 \beta - 6) q^{43} + (6 \beta + 6) q^{44} - 10 q^{46} + (2 \beta - 8) q^{47} - 2 q^{48} + ( - 4 \beta + 1) q^{49} + ( - 2 \beta + 2) q^{51} + ( - 9 \beta + 9) q^{52} + ( - 5 \beta + 8) q^{53} + (8 \beta - 4) q^{54} + ( - 2 \beta + 6) q^{56} + (3 \beta + 11) q^{58} - 2 \beta q^{59} + (3 \beta + 5) q^{61} + ( - 12 \beta + 6) q^{62} + ( - 2 \beta + 2) q^{63} - 13 q^{64} + ( - 12 \beta - 4) q^{66} + ( - 4 \beta + 6) q^{67} + ( - 3 \beta + 3) q^{68} + (8 \beta - 4) q^{69} + 2 \beta q^{71} + ( - 2 \beta + 1) q^{72} + 9 \beta q^{73} + ( - 11 \beta - 2) q^{74} - 4 \beta q^{77} + ( - 6 \beta + 18) q^{78} - 2 q^{79} - 11 q^{81} + ( - 3 \beta + 14) q^{82} + 6 \beta q^{83} + ( - 12 \beta + 12) q^{84} + (8 \beta - 14) q^{86} + ( - 10 \beta + 2) q^{87} + ( - 6 \beta - 2) q^{88} + (7 \beta - 4) q^{89} + ( - 6 \beta + 12) q^{91} + (12 \beta - 6) q^{92} + 12 q^{93} + (14 \beta - 12) q^{94} + (12 \beta - 6) q^{96} + ( - 5 \beta - 3) q^{97} + (2 \beta + 9) q^{98} + (2 \beta + 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{3} + 6 q^{4} + 2 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{3} + 6 q^{4} + 2 q^{7} + 2 q^{9} + 6 q^{11} + 12 q^{12} + 3 q^{13} + 10 q^{14} - 2 q^{16} + q^{17} + 4 q^{21} - 10 q^{22} + 15 q^{26} - 8 q^{27} + 6 q^{28} - 3 q^{29} + 12 q^{31} + 12 q^{33} + 5 q^{34} + 6 q^{36} + 11 q^{37} + 6 q^{39} + 3 q^{41} + 20 q^{42} - 8 q^{43} + 18 q^{44} - 20 q^{46} - 14 q^{47} - 4 q^{48} - 2 q^{49} + 2 q^{51} + 9 q^{52} + 11 q^{53} + 10 q^{56} + 25 q^{58} - 2 q^{59} + 13 q^{61} + 2 q^{63} - 26 q^{64} - 20 q^{66} + 8 q^{67} + 3 q^{68} + 2 q^{71} + 9 q^{73} - 15 q^{74} - 4 q^{77} + 30 q^{78} - 4 q^{79} - 22 q^{81} + 25 q^{82} + 6 q^{83} + 12 q^{84} - 20 q^{86} - 6 q^{87} - 10 q^{88} - q^{89} + 18 q^{91} + 24 q^{93} - 10 q^{94} - 11 q^{97} + 20 q^{98} + 6 q^{99}+O(q^{100}) \) 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
−2.23607 2.00000 3.00000 0 −4.47214 −1.23607 −2.23607 1.00000 0
1.2 2.23607 2.00000 3.00000 0 4.47214 3.23607 2.23607 1.00000 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.r 2
5.b even 2 1 361.2.a.d 2
15.d odd 2 1 3249.2.a.n 2
19.b odd 2 1 9025.2.a.o 2
20.d odd 2 1 5776.2.a.bh 2
95.d odd 2 1 361.2.a.e yes 2
95.h odd 6 2 361.2.c.e 4
95.i even 6 2 361.2.c.f 4
95.o odd 18 6 361.2.e.k 12
95.p even 18 6 361.2.e.l 12
285.b even 2 1 3249.2.a.m 2
380.d even 2 1 5776.2.a.r 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
361.2.a.d 2 5.b even 2 1
361.2.a.e yes 2 95.d odd 2 1
361.2.c.e 4 95.h odd 6 2
361.2.c.f 4 95.i even 6 2
361.2.e.k 12 95.o odd 18 6
361.2.e.l 12 95.p even 18 6
3249.2.a.m 2 285.b even 2 1
3249.2.a.n 2 15.d odd 2 1
5776.2.a.r 2 380.d even 2 1
5776.2.a.bh 2 20.d odd 2 1
9025.2.a.o 2 19.b odd 2 1
9025.2.a.r 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9025))\):

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 5 \) Copy content Toggle raw display
$3$ \( (T - 2)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$11$ \( T^{2} - 6T + 4 \) Copy content Toggle raw display
$13$ \( T^{2} - 3T - 9 \) Copy content Toggle raw display
$17$ \( T^{2} - T - 1 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 20 \) Copy content Toggle raw display
$29$ \( T^{2} + 3T - 29 \) Copy content Toggle raw display
$31$ \( (T - 6)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} - 11T + 19 \) Copy content Toggle raw display
$41$ \( T^{2} - 3T - 29 \) Copy content Toggle raw display
$43$ \( T^{2} + 8T - 4 \) Copy content Toggle raw display
$47$ \( T^{2} + 14T + 44 \) Copy content Toggle raw display
$53$ \( T^{2} - 11T - 1 \) Copy content Toggle raw display
$59$ \( T^{2} + 2T - 4 \) Copy content Toggle raw display
$61$ \( T^{2} - 13T + 31 \) Copy content Toggle raw display
$67$ \( T^{2} - 8T - 4 \) Copy content Toggle raw display
$71$ \( T^{2} - 2T - 4 \) Copy content Toggle raw display
$73$ \( T^{2} - 9T - 81 \) Copy content Toggle raw display
$79$ \( (T + 2)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} - 6T - 36 \) Copy content Toggle raw display
$89$ \( T^{2} + T - 61 \) Copy content Toggle raw display
$97$ \( T^{2} + 11T - 1 \) Copy content Toggle raw display
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