Properties

Label 1225.2.a.bc
Level $1225$
Weight $2$
Character orbit 1225.a
Self dual yes
Analytic conductor $9.782$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1225,2,Mod(1,1225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1225, 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("1225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1225.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: \(9.78167424761\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 6x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 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.

\(f(q)\) \(=\) \( q + (\beta_{2} + 1) q^{2} + \beta_1 q^{3} + 3 \beta_{2} q^{4} + (\beta_{3} + 2 \beta_1) q^{6} + (4 \beta_{2} + 1) q^{8} + (2 \beta_{2} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 1) q^{2} + \beta_1 q^{3} + 3 \beta_{2} q^{4} + (\beta_{3} + 2 \beta_1) q^{6} + (4 \beta_{2} + 1) q^{8} + (2 \beta_{2} - 1) q^{9} + ( - 2 \beta_{2} + 1) q^{11} + (3 \beta_{3} + 3 \beta_1) q^{12} + (\beta_{3} - 3 \beta_1) q^{13} + (3 \beta_{2} + 5) q^{16} + ( - 4 \beta_{3} + 2 \beta_1) q^{17} + (3 \beta_{2} + 1) q^{18} - 3 \beta_{3} q^{19} + ( - 3 \beta_{2} - 1) q^{22} + (2 \beta_{2} + 5) q^{23} + (4 \beta_{3} + 5 \beta_1) q^{24} + ( - 2 \beta_{3} - 5 \beta_1) q^{26} + (2 \beta_{3} - 2 \beta_1) q^{27} + ( - 2 \beta_{2} + 1) q^{29} + ( - 3 \beta_{3} + 3 \beta_1) q^{31} + (3 \beta_{2} + 6) q^{32} + ( - 2 \beta_{3} - \beta_1) q^{33} - 2 \beta_{3} q^{34} + (3 \beta_{2} + 6) q^{36} + ( - 6 \beta_{2} + 7) q^{37} + ( - 3 \beta_{3} - 3 \beta_1) q^{38} + ( - 4 \beta_{2} - 6) q^{39} + (4 \beta_{3} + \beta_1) q^{41} + 5 q^{43} + ( - 3 \beta_{2} - 6) q^{44} + (9 \beta_{2} + 7) q^{46} + (3 \beta_{3} - 6 \beta_1) q^{47} + (3 \beta_{3} + 8 \beta_1) q^{48} + ( - 4 \beta_{2} + 4) q^{51} + ( - 9 \beta_{3} - 6 \beta_1) q^{52} + ( - 6 \beta_{2} + 6) q^{53} - 2 \beta_1 q^{54} - 6 \beta_{2} q^{57} + ( - 3 \beta_{2} - 1) q^{58} + (5 \beta_{3} - 7 \beta_1) q^{59} + (4 \beta_{3} + 3 \beta_1) q^{61} + 3 \beta_1 q^{62} + (6 \beta_{2} - 1) q^{64} + ( - 3 \beta_{3} - 4 \beta_1) q^{66} + ( - 6 \beta_{2} + 5) q^{67} + (6 \beta_{3} - 6 \beta_1) q^{68} + (2 \beta_{3} + 7 \beta_1) q^{69} + (4 \beta_{2} - 5) q^{71} + (6 \beta_{2} + 7) q^{72} + 3 \beta_1 q^{73} + ( - 5 \beta_{2} + 1) q^{74} - 9 \beta_1 q^{76} + ( - 14 \beta_{2} - 10) q^{78} + (6 \beta_{2} - 1) q^{79} + ( - 6 \beta_{2} - 1) q^{81} + (5 \beta_{3} + 6 \beta_1) q^{82} + ( - 3 \beta_{3} + 3 \beta_1) q^{83} + (5 \beta_{2} + 5) q^{86} + ( - 2 \beta_{3} - \beta_1) q^{87} + ( - 6 \beta_{2} - 7) q^{88} + ( - 5 \beta_{3} + \beta_1) q^{89} + (21 \beta_{2} + 6) q^{92} + 6 q^{93} + ( - 3 \beta_{3} - 9 \beta_1) q^{94} + (3 \beta_{3} + 9 \beta_1) q^{96} + (9 \beta_{3} + 3 \beta_1) q^{97} - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 6 q^{2} + 6 q^{4} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 6 q^{2} + 6 q^{4} + 12 q^{8} + 26 q^{16} + 10 q^{18} - 10 q^{22} + 24 q^{23} + 30 q^{32} + 30 q^{36} + 16 q^{37} - 32 q^{39} + 20 q^{43} - 30 q^{44} + 46 q^{46} + 8 q^{51} + 12 q^{53} - 12 q^{57} - 10 q^{58} + 8 q^{64} + 8 q^{67} - 12 q^{71} + 40 q^{72} - 6 q^{74} - 68 q^{78} + 8 q^{79} - 16 q^{81} + 30 q^{86} - 40 q^{88} + 66 q^{92} + 24 q^{93} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 6x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - 2 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 4\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + 4\beta_1 \) 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
−0.874032
0.874032
−2.28825
2.28825
0.381966 −0.874032 −1.85410 0 −0.333851 0 −1.47214 −2.23607 0
1.2 0.381966 0.874032 −1.85410 0 0.333851 0 −1.47214 −2.23607 0
1.3 2.61803 −2.28825 4.85410 0 −5.99070 0 7.47214 2.23607 0
1.4 2.61803 2.28825 4.85410 0 5.99070 0 7.47214 2.23607 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(7\) \(1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1225.2.a.bc yes 4
5.b even 2 1 1225.2.a.ba 4
5.c odd 4 2 1225.2.b.n 8
7.b odd 2 1 inner 1225.2.a.bc yes 4
35.c odd 2 1 1225.2.a.ba 4
35.f even 4 2 1225.2.b.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1225.2.a.ba 4 5.b even 2 1
1225.2.a.ba 4 35.c odd 2 1
1225.2.a.bc yes 4 1.a even 1 1 trivial
1225.2.a.bc yes 4 7.b odd 2 1 inner
1225.2.b.n 8 5.c odd 4 2
1225.2.b.n 8 35.f even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1225))\):

\( T_{2}^{2} - 3T_{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{4} - 6T_{3}^{2} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 3 T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - 6T^{2} + 4 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} - 5)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 46T^{2} + 484 \) Copy content Toggle raw display
$17$ \( T^{4} - 56T^{2} + 64 \) Copy content Toggle raw display
$19$ \( (T^{2} - 18)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 12 T + 31)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 5)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 54T^{2} + 324 \) Copy content Toggle raw display
$37$ \( (T^{2} - 8 T - 29)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 86T^{2} + 1444 \) Copy content Toggle raw display
$43$ \( (T - 5)^{4} \) Copy content Toggle raw display
$47$ \( (T^{2} - 90)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} - 6 T - 36)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 254 T^{2} + 13924 \) Copy content Toggle raw display
$61$ \( T^{4} - 166T^{2} + 1444 \) Copy content Toggle raw display
$67$ \( (T^{2} - 4 T - 41)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 6 T - 11)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} - 54T^{2} + 324 \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T - 41)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 54T^{2} + 324 \) Copy content Toggle raw display
$89$ \( T^{4} - 86T^{2} + 1444 \) Copy content Toggle raw display
$97$ \( T^{4} - 486 T^{2} + 39204 \) Copy content Toggle raw display
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