↑

Properties

Label 4225.2.a.bj
Level $4225$
Weight $2$
Character orbit 4225.a
Self dual yes
Analytic conductor $33.737$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4225,2,Mod(1,4225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4225, 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("4225.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4225 = 5^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4225.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: \(33.7367948540\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{3}, \sqrt{7})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 5x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} - q^{3} - \beta_{2} q^{4} - \beta_{3} q^{6} + ( - \beta_{3} - \beta_1) q^{7} + (\beta_{3} + \beta_1) q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} - q^{3} - \beta_{2} q^{4} - \beta_{3} q^{6} + ( - \beta_{3} - \beta_1) q^{7} + (\beta_{3} + \beta_1) q^{8} - 2 q^{9} + (\beta_{3} - \beta_1) q^{11} + \beta_{2} q^{12} + (\beta_{2} - 1) q^{14} + (\beta_{2} + 1) q^{16} + ( - 2 \beta_{2} - 1) q^{17} - 2 \beta_{3} q^{18} + (\beta_{3} + \beta_1) q^{19} + (\beta_{3} + \beta_1) q^{21} + ( - \beta_{2} + 3) q^{22} + (2 \beta_{2} + 1) q^{23} + ( - \beta_{3} - \beta_1) q^{24} + 5 q^{27} + ( - 2 \beta_{3} + \beta_1) q^{28} + (2 \beta_{2} + 1) q^{29} + ( - 4 \beta_{3} + 2 \beta_1) q^{31} + ( - 4 \beta_{3} - 3 \beta_1) q^{32} + ( - \beta_{3} + \beta_1) q^{33} + (5 \beta_{3} + 2 \beta_1) q^{34} + 2 \beta_{2} q^{36} + ( - 3 \beta_{3} + 3 \beta_1) q^{37} + ( - \beta_{2} + 1) q^{38} + (\beta_{3} - \beta_1) q^{41} + ( - \beta_{2} + 1) q^{42} + (2 \beta_{2} + 7) q^{43} + (4 \beta_{3} + 3 \beta_1) q^{44} + ( - 5 \beta_{3} - 2 \beta_1) q^{46} + 4 \beta_{3} q^{47} + ( - \beta_{2} - 1) q^{48} - 4 q^{49} + (2 \beta_{2} + 1) q^{51} + ( - 2 \beta_{2} - 4) q^{53} + 5 \beta_{3} q^{54} - 3 q^{56} + ( - \beta_{3} - \beta_1) q^{57} + ( - 5 \beta_{3} - 2 \beta_1) q^{58} + ( - 3 \beta_{3} - 7 \beta_1) q^{59} + (2 \beta_{2} - 5) q^{61} + (4 \beta_{2} - 10) q^{62} + (2 \beta_{3} + 2 \beta_1) q^{63} + (2 \beta_{2} - 7) q^{64} + (\beta_{2} - 3) q^{66} + ( - 7 \beta_{3} - \beta_1) q^{67} + ( - \beta_{2} + 10) q^{68} + ( - 2 \beta_{2} - 1) q^{69} + ( - \beta_{3} + 3 \beta_1) q^{71} + ( - 2 \beta_{3} - 2 \beta_1) q^{72} + (3 \beta_{2} - 9) q^{74} + (2 \beta_{3} - \beta_1) q^{76} + (2 \beta_{2} + 1) q^{77} - 6 q^{79} + q^{81} + ( - \beta_{2} + 3) q^{82} + (6 \beta_{3} + 4 \beta_1) q^{83} + (2 \beta_{3} - \beta_1) q^{84} + (\beta_{3} - 2 \beta_1) q^{86} + ( - 2 \beta_{2} - 1) q^{87} + ( - 2 \beta_{2} - 1) q^{88} + ( - 3 \beta_{3} - 5 \beta_1) q^{89} + (\beta_{2} - 10) q^{92} + (4 \beta_{3} - 2 \beta_1) q^{93} + ( - 4 \beta_{2} + 8) q^{94} + (4 \beta_{3} + 3 \beta_1) q^{96} + ( - \beta_{3} + 5 \beta_1) q^{97} - 4 \beta_{3} q^{98} + ( - 2 \beta_{3} + 2 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} + 2 q^{4} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} + 2 q^{4} - 8 q^{9} - 2 q^{12} - 6 q^{14} + 2 q^{16} + 14 q^{22} + 20 q^{27} - 4 q^{36} + 6 q^{38} + 6 q^{42} + 24 q^{43} - 2 q^{48} - 16 q^{49} - 12 q^{53} - 12 q^{56} - 24 q^{61} - 48 q^{62} - 32 q^{64} - 14 q^{66} + 42 q^{68} - 42 q^{74} - 24 q^{79} + 4 q^{81} + 14 q^{82} - 42 q^{92} + 40 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 5x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
0.456850
2.18890
−2.18890
−0.456850
−2.18890 −1.00000 2.79129 0 2.18890 1.73205 −1.73205 −2.00000 0
1.2 −0.456850 −1.00000 −1.79129 0 0.456850 −1.73205 1.73205 −2.00000 0
1.3 0.456850 −1.00000 −1.79129 0 −0.456850 1.73205 −1.73205 −2.00000 0
1.4 2.18890 −1.00000 2.79129 0 −2.18890 −1.73205 1.73205 −2.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \(-1\)
\(13\) \(-1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4225.2.a.bj 4
5.b even 2 1 4225.2.a.bk 4
5.c odd 4 2 845.2.b.f 8
13.b even 2 1 inner 4225.2.a.bj 4
13.f odd 12 2 325.2.n.b 4
65.d even 2 1 4225.2.a.bk 4
65.f even 4 2 845.2.d.c 8
65.h odd 4 2 845.2.b.f 8
65.k even 4 2 845.2.d.c 8
65.o even 12 2 65.2.l.a ✓ 8
65.o even 12 2 845.2.l.c 8
65.q odd 12 2 845.2.n.c 8
65.q odd 12 2 845.2.n.d 8
65.r odd 12 2 845.2.n.c 8
65.r odd 12 2 845.2.n.d 8
65.s odd 12 2 325.2.n.c 4
65.t even 12 2 65.2.l.a ✓ 8
65.t even 12 2 845.2.l.c 8
195.bc odd 12 2 585.2.bf.a 8
195.bn odd 12 2 585.2.bf.a 8
260.be odd 12 2 1040.2.df.b 8
260.bl odd 12 2 1040.2.df.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.l.a ✓ 8 65.o even 12 2
65.2.l.a ✓ 8 65.t even 12 2
325.2.n.b 4 13.f odd 12 2
325.2.n.c 4 65.s odd 12 2
585.2.bf.a 8 195.bc odd 12 2
585.2.bf.a 8 195.bn odd 12 2
845.2.b.f 8 5.c odd 4 2
845.2.b.f 8 65.h odd 4 2
845.2.d.c 8 65.f even 4 2
845.2.d.c 8 65.k even 4 2
845.2.l.c 8 65.o even 12 2
845.2.l.c 8 65.t even 12 2
845.2.n.c 8 65.q odd 12 2
845.2.n.c 8 65.r odd 12 2
845.2.n.d 8 65.q odd 12 2
845.2.n.d 8 65.r odd 12 2
1040.2.df.b 8 260.be odd 12 2
1040.2.df.b 8 260.bl odd 12 2
4225.2.a.bj 4 1.a even 1 1 trivial
4225.2.a.bj 4 13.b even 2 1 inner
4225.2.a.bk 4 5.b even 2 1
4225.2.a.bk 4 65.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4225))\):

\( T_{2}^{4} - 5T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} - 3 \) Copy content Toggle raw display
\( T_{11}^{2} - 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 5T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 7)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} \) Copy content Toggle raw display
$17$ \( (T^{2} - 21)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$23$ \( (T^{2} - 21)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 21)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} - 132T^{2} + 3600 \) Copy content Toggle raw display
$37$ \( (T^{2} - 63)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 7)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} - 12 T + 15)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} - 80T^{2} + 256 \) Copy content Toggle raw display
$53$ \( (T^{2} + 6 T - 12)^{2} \) Copy content Toggle raw display
$59$ \( T^{4} - 206T^{2} + 2209 \) Copy content Toggle raw display
$61$ \( (T^{2} + 12 T + 15)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 222T^{2} + 225 \) Copy content Toggle raw display
$71$ \( T^{4} - 62T^{2} + 625 \) Copy content Toggle raw display
$73$ \( T^{4} \) Copy content Toggle raw display
$79$ \( (T + 6)^{4} \) Copy content Toggle raw display
$83$ \( T^{4} - 164T^{2} + 4624 \) Copy content Toggle raw display
$89$ \( T^{4} - 110T^{2} + 1681 \) Copy content Toggle raw display
$97$ \( T^{4} - 150T^{2} + 2601 \) Copy content Toggle raw display
show more
show less