Properties

Label 1225.4.a.u
Level $1225$
Weight $4$
Character orbit 1225.a
Self dual yes
Analytic conductor $72.277$
Analytic rank $0$
Dimension $2$
CM discriminant -7
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1225,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1225.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(72.2773397570\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{21}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 5 \) 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: $N(\mathrm{U}(1))$

$q$-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 \(\beta = \frac{1}{2}(1 + \sqrt{21})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta + 3) q^{2} + ( - 5 \beta + 6) q^{4} + ( - 8 \beta + 19) q^{8} - 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta + 3) q^{2} + ( - 5 \beta + 6) q^{4} + ( - 8 \beta + 19) q^{8} - 27 q^{9} + ( - 10 \beta + 39) q^{11} + (5 \beta + 49) q^{16} + (27 \beta - 81) q^{18} + ( - 59 \beta + 167) q^{22} + ( - 82 \beta + 61) q^{23} + ( - 100 \beta + 133) q^{29} + (25 \beta - 30) q^{32} + (135 \beta - 162) q^{36} + ( - 4 \beta - 223) q^{37} + ( - 202 \beta + 191) q^{43} + ( - 205 \beta + 484) q^{44} + ( - 225 \beta + 593) q^{46} + 590 q^{53} + ( - 333 \beta + 899) q^{58} + (40 \beta - 607) q^{64} + (306 \beta + 217) q^{67} + (370 \beta - 529) q^{71} + (216 \beta - 513) q^{72} + (215 \beta - 649) q^{74} + ( - 90 \beta + 737) q^{79} + 729 q^{81} + ( - 595 \beta + 1583) q^{86} + ( - 422 \beta + 1141) q^{88} + ( - 387 \beta + 2416) q^{92} + (270 \beta - 1053) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 5 q^{2} + 7 q^{4} + 30 q^{8} - 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 5 q^{2} + 7 q^{4} + 30 q^{8} - 54 q^{9} + 68 q^{11} + 103 q^{16} - 135 q^{18} + 275 q^{22} + 40 q^{23} + 166 q^{29} - 35 q^{32} - 189 q^{36} - 450 q^{37} + 180 q^{43} + 763 q^{44} + 961 q^{46} + 1180 q^{53} + 1465 q^{58} - 1174 q^{64} + 740 q^{67} - 688 q^{71} - 810 q^{72} - 1083 q^{74} + 1384 q^{79} + 1458 q^{81} + 2571 q^{86} + 1860 q^{88} + 4445 q^{92} - 1836 q^{99}+O(q^{100}) \) 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
2.79129
−1.79129
0.208712 0 −7.95644 0 0 0 −3.33030 −27.0000 0
1.2 4.79129 0 14.9564 0 0 0 33.3303 −27.0000 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 CM by \(\Q(\sqrt{-7}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1225.4.a.u yes 2
5.b even 2 1 1225.4.a.o 2
7.b odd 2 1 CM 1225.4.a.u yes 2
35.c odd 2 1 1225.4.a.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1225.4.a.o 2 5.b even 2 1
1225.4.a.o 2 35.c odd 2 1
1225.4.a.u yes 2 1.a even 1 1 trivial
1225.4.a.u yes 2 7.b odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1225))\):

\( T_{2}^{2} - 5T_{2} + 1 \) Copy content Toggle raw display
\( T_{3} \) Copy content Toggle raw display
\( T_{19} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - 5T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} \) Copy content Toggle raw display
$11$ \( T^{2} - 68T + 631 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} - 40T - 34901 \) Copy content Toggle raw display
$29$ \( T^{2} - 166T - 45611 \) Copy content Toggle raw display
$31$ \( T^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 450T + 50541 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 180T - 206121 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T - 590)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} \) Copy content Toggle raw display
$67$ \( T^{2} - 740T - 354689 \) Copy content Toggle raw display
$71$ \( T^{2} + 688T - 600389 \) Copy content Toggle raw display
$73$ \( T^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 1384 T + 436339 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} \) Copy content Toggle raw display
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