Properties

Label 9025.2.a.bg
Level $9025$
Weight $2$
Character orbit 9025.a
Self dual yes
Analytic conductor $72.065$
Analytic rank $1$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.7537.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 5x^{2} + 4x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + (\beta_1 - 1) q^{3} + (\beta_{3} - \beta_{2} + 1) q^{4} + (\beta_{3} - \beta_{2} + \beta_1) q^{6} + (\beta_{3} + 1) q^{7} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 - 3) q^{8} + (\beta_{2} - 2 \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + (\beta_1 - 1) q^{3} + (\beta_{3} - \beta_{2} + 1) q^{4} + (\beta_{3} - \beta_{2} + \beta_1) q^{6} + (\beta_{3} + 1) q^{7} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 - 3) q^{8} + (\beta_{2} - 2 \beta_1 + 1) q^{9} + ( - \beta_{3} - 2 \beta_1) q^{11} + ( - \beta_{3} + 2 \beta_{2} - 1) q^{12} + (\beta_{3} + \beta_1 - 2) q^{13} + ( - \beta_{3} + 2 \beta_{2} + \beta_1) q^{14} + (2 \beta_{3} - 4 \beta_{2} - \beta_1 + 1) q^{16} + (\beta_{2} + 2 \beta_1) q^{17} + ( - \beta_{3} - 2 \beta_1 + 3) q^{18} + (\beta_{2} + \beta_1 - 1) q^{21} + ( - \beta_{3} - \beta_{2} - 3 \beta_1) q^{22} + (\beta_{3} + \beta_{2} - \beta_1) q^{23} + (\beta_{3} - 2 \beta_{2} - 3 \beta_1 + 6) q^{24} + ( - \beta_{2} + 2 \beta_1) q^{26} + (\beta_{3} - 3 \beta_{2} + \beta_1 - 4) q^{27} + (2 \beta_{3} - 3 \beta_{2} + 4) q^{28} + ( - 2 \beta_{3} + 2 \beta_{2} + \beta_1) q^{29} + ( - \beta_{3} - 3 \beta_{2} + \beta_1 - 1) q^{31} + ( - 3 \beta_{3} + 5 \beta_{2} + \cdots - 6) q^{32}+ \cdots + (5 \beta_{2} - 5 \beta_1 + 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} - 3 q^{3} + 5 q^{4} + 2 q^{6} + 4 q^{7} - 12 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} - 3 q^{3} + 5 q^{4} + 2 q^{6} + 4 q^{7} - 12 q^{8} + q^{9} - 2 q^{11} - 6 q^{12} - 7 q^{13} - q^{14} + 7 q^{16} + q^{17} + 10 q^{18} - 4 q^{21} - 2 q^{22} - 2 q^{23} + 23 q^{24} + 3 q^{26} - 12 q^{27} + 19 q^{28} - q^{29} - 30 q^{32} - 19 q^{33} + 15 q^{34} - 7 q^{36} + 2 q^{37} + 15 q^{39} - 8 q^{41} + 15 q^{42} - q^{43} - 12 q^{44} + 12 q^{46} + 12 q^{47} - 23 q^{48} - 10 q^{49} + 22 q^{51} + 3 q^{52} + 5 q^{53} - 34 q^{54} - 41 q^{56} + 27 q^{58} - 5 q^{59} - 37 q^{62} + 3 q^{63} + 56 q^{64} - 31 q^{66} - 4 q^{67} - 16 q^{68} - 9 q^{69} + 20 q^{71} - 17 q^{72} + 20 q^{73} + 25 q^{74} - 14 q^{77} + 18 q^{78} + 17 q^{79} + 12 q^{81} - 21 q^{82} - q^{83} - 20 q^{84} + 8 q^{86} + 16 q^{87} + 7 q^{88} + 11 q^{89} + 6 q^{91} + q^{92} + 8 q^{93} - 31 q^{94} + 21 q^{96} - q^{97} - 9 q^{98} + 38 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 5x^{2} + 4x + 3 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta_1 \) 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
−0.491918
1.37933
−2.04717
2.15976
−2.75802 −1.49192 5.60665 0 4.11474 2.84864 −9.94721 −0.774179 0
1.2 −1.09744 0.379334 −0.795629 0 −0.416295 −1.89307 3.06803 −2.85611 0
1.3 1.19091 −3.04717 −0.581734 0 −3.62891 0.609175 −3.07461 6.28525 0
1.4 1.66454 1.15976 0.770710 0 1.93047 2.43525 −2.04621 −1.65497 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.bg 4
5.b even 2 1 1805.2.a.o 4
19.b odd 2 1 9025.2.a.bp 4
19.c even 3 2 475.2.e.e 8
95.d odd 2 1 1805.2.a.i 4
95.i even 6 2 95.2.e.c 8
95.m odd 12 4 475.2.j.c 16
285.n odd 6 2 855.2.k.h 8
380.p odd 6 2 1520.2.q.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.e.c 8 95.i even 6 2
475.2.e.e 8 19.c even 3 2
475.2.j.c 16 95.m odd 12 4
855.2.k.h 8 285.n odd 6 2
1520.2.q.o 8 380.p odd 6 2
1805.2.a.i 4 95.d odd 2 1
1805.2.a.o 4 5.b even 2 1
9025.2.a.bg 4 1.a even 1 1 trivial
9025.2.a.bp 4 19.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9025))\):

\( T_{2}^{4} + T_{2}^{3} - 6T_{2}^{2} - T_{2} + 6 \) Copy content Toggle raw display
\( T_{3}^{4} + 3T_{3}^{3} - 2T_{3}^{2} - 5T_{3} + 2 \) Copy content Toggle raw display
\( T_{7}^{4} - 4T_{7}^{3} - T_{7}^{2} + 15T_{7} - 8 \) Copy content Toggle raw display
\( T_{11}^{4} + 2T_{11}^{3} - 25T_{11}^{2} - 19T_{11} + 3 \) Copy content Toggle raw display
\( T_{29}^{4} + T_{29}^{3} - 63T_{29}^{2} + 194T_{29} - 141 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} - 6 T^{2} + \cdots + 6 \) Copy content Toggle raw display
$3$ \( T^{4} + 3 T^{3} + \cdots + 2 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 4 T^{3} + \cdots - 8 \) Copy content Toggle raw display
$11$ \( T^{4} + 2 T^{3} + \cdots + 3 \) Copy content Toggle raw display
$13$ \( T^{4} + 7 T^{3} + \cdots - 16 \) Copy content Toggle raw display
$17$ \( T^{4} - T^{3} + \cdots + 108 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 2 T^{3} + \cdots + 6 \) Copy content Toggle raw display
$29$ \( T^{4} + T^{3} + \cdots - 141 \) Copy content Toggle raw display
$31$ \( T^{4} - 67 T^{2} + \cdots + 1063 \) Copy content Toggle raw display
$37$ \( T^{4} - 2 T^{3} + \cdots - 118 \) Copy content Toggle raw display
$41$ \( T^{4} + 8 T^{3} + \cdots - 2238 \) Copy content Toggle raw display
$43$ \( T^{4} + T^{3} + \cdots + 794 \) Copy content Toggle raw display
$47$ \( T^{4} - 12 T^{3} + \cdots - 2316 \) Copy content Toggle raw display
$53$ \( T^{4} - 5 T^{3} + \cdots - 54 \) Copy content Toggle raw display
$59$ \( T^{4} + 5 T^{3} + \cdots + 1875 \) Copy content Toggle raw display
$61$ \( T^{4} - 130 T^{2} + \cdots + 3049 \) Copy content Toggle raw display
$67$ \( T^{4} + 4 T^{3} + \cdots + 64 \) Copy content Toggle raw display
$71$ \( T^{4} - 20 T^{3} + \cdots - 243 \) Copy content Toggle raw display
$73$ \( T^{4} - 20 T^{3} + \cdots - 1726 \) Copy content Toggle raw display
$79$ \( T^{4} - 17 T^{3} + \cdots - 184 \) Copy content Toggle raw display
$83$ \( T^{4} + T^{3} + \cdots + 366 \) Copy content Toggle raw display
$89$ \( T^{4} - 11 T^{3} + \cdots - 3816 \) Copy content Toggle raw display
$97$ \( T^{4} + T^{3} + \cdots + 7442 \) Copy content Toggle raw display
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